You would have to get a GC out to inspect the attic and then give you a written recommendation of alterations, which you then would take to your association. Talk with your neighbors and see if they have the same issue, them jumping in on the issue would offer a greater chance of approval. It is possible the association may pay for it if the problem is attic venting, insulation would likely fall on you.
Forcing the system to run more to compensate for the upstairs would drive up your electric bill and may not actually help if the problem is in the attic or balance of the HVAC. You could set the temp lower at night to accomplish the same thing as dual sensors, but you still are just throwing money away that should be better spent on addressing the issue directly.
Having a nest vs non-smart thermostat allows you to set schedules in a smarter way than a regular tstat.
Having dampers allows you to balance the system so the downstairs gets less air than the upstairs, instead of you having to go around closing vents and the downstairs would be cooled better than just shutting vents and sending most of the air upstairs.
The mini-split route idea is a great solution, but I am wondering if you have an unvented or poorly insulated attic that is a contributing factor to your problem. If your attic space isn't setup properly and you throw a mini-split at the problem, it is sort of like pumping water out of a boat without fixing the leak that is flooding the boat. Even something as "simple" as a powered attic vent could help in HVAC performance and operating costs. If the hot air can't vent or your insulation is inadequate, it just stacks down and heats the top floor.
The other thing I would be looking at is if the HVAC run up to the second floor is done properly or if the dampers are not right. A house that size really shouldn't have much trouble if everything is balanced right.
Closing vents on the first floor and leaving the second floor open will increase the wear on the fan and may decrease efficiency if not enough air can be moved through the open vents. Knowing how bad an idea that is would require some measurements and math.
Having the nest with a second temp sensor would just force your HVAC to work harder to cool the upstairs. The single sensor nest can be a smart move to reduce costs with the higher degree of control.
Having dampers installed does more or less the same thing as closing vents, but with a higher cost. A zoned system is expensive and kind of silly on a house of that size.
The flange they adapted was done pretty badly so it isn't making as even contact as would be ideal. Looks like they tried to use sealant to make up for it and didn't use enough, so it still had gaps.
If it were my circus, I'd use butyl sheet under the boot for more expansion/contraction tolerant waterproofing and take the extra time to do a better job on that flange.
You could go ham with some roof patch and be done with it for a decade and do it better when it needs redoing.
I would use a grinder to put rebar down over the gap, set into the bricks, put cement board on top of that which has a 1/2" overhang onto the brick. I am extra and would do a rebar every brick across the short side. From there you can built a mold to pour a rebar and concrete slab with a 2" overhang, or no overhang if your rain cap can accommodate the tile extensions. The overhang will help to protect the mortar joints of the brick, so if your cap has less drip edge, I would just pour with the overhang. You do want to make sure the concrete is shaped to allow for water runoff, water sitting on the crown will fuck shit up faster if it sits. Use some foam seal around the tiles to give some relief for expansion and contraction. Seal over the foam with an exterior polyurethane sealant so water will have a harder time acting on the joint between the tile and the concrete.
If you elect to have the tiles with their own individual rain caps, I would suggest adding a sand cap over the concrete to act as a sacrificial wash. You'll want to slip a tuck pointing trowel up against the tile to give some expansion room. Seal that wash to the tile with polyurethane as well. You don't need more than an inch thickness, tapering to 1/2" at the edge. You will have to replace/patch that sand cap every so often, so make sure to check on it. Most of the time you just need to patch it with the leftover sand cap thinned out and applied with a brush, but you can use mortar as well.
So it is basically a closed chase with brick surrounding the flue. That isn't how it would be built today, but very common on most older homes. What you really don't want is naked flue tiles in an open chase with no brick supporting them because they will be prone to serious failures like the tile stack collapsing.
You don't really need to worry about that cavity. They want the brick cores filled and adequate mortar joints in the brick, not dry set(no mortar).
Construct a new crown to seal off the chase and extend the flue tiles by about 6" off the top of the new crown. There is a chance of a cross breeze pulling smoke down the adjacent flue, but there are a few factors that can cause that and I would need a few photos of the whole picture to have a decent idea if that could be an issue.
Does the wall dividing the gap and the flue tiles visible from the top go all the way down or does it stop after a few bricks and then just has the flue stack naked in the chase?
I'm certain neither flue would meet modern safety standards, and that is just from what I can see in that picture with very little information. There are some grandfathering of some issues.
Both flue almost certainly has deteriorated mortar joints or damaged tiles. You would need to have the flue camera inspected(level 2 inspection).
The right flue is fully fucked and should not be used. The tiles are gone and there is masonry that has fallen and is obstructing the flue. Edit: Not a flue, is open chase cavity.
To determine if the flue will be able to have a stainless liner, you need the measurements of the firebox width, depth, and rear wall width. You then can put those measurements into an online calculator to determine liner size. The next problem is how much space you have in the chase(chimney structure surrounding the liner). If your chase has an opening of 10"x10" and you need an 11" liner, that is a problem.
The flue may be able to be coated with a masonry product like Fireguard or Fluesaver. That would depend on the condition of the flue tiles and the exact system used to reline.
The top of the chimney should not be closed off with a cap like you have there. It should have the tiles or a liner extending past the crown(top) of the chimney, the crown would close off the chase and keep water, debris, animals, etc. from being able to enter the chase. Water and products of combustion acting on the mortar joints of tiles and brick will destroy the structure. Just putting that cap on there is better than nothing, but it is more of a way to buy time so you can actually fix it properly. The crown used to be covered and sealed properly, but has deteriorated to the state it is in now.
Get a certified chimney sweep out there for a level 2 inspection of both flue and they will be able to give you a better understanding of what all needs to be done and you can go from there. Use a company that does the inspections and the work. If they don't do the work, then you will have to deal with inaccurate quotes and a bigger headache. Just do an inspection on the right one, no sweep. They should be swept with the inspection to get a good idea of the tile and mortar joint condition.
Edit: fixed some stuff based on further explanation of the situation.
The left side flue was appropriate when it was built with a closed chase supporting the flue tiles. The right side has no flue tiles and would only be acceptable like a 100 years ago.
Epoxy or silicone, I favor epoxy. You will need to make sure everything is absolutely clean and you will probably want to take some sandpaper to them to provide a better mechanical grip.
When you say flexible ducting, are you talking aluminized plastic or the corrugated aluminum? That would be a fair amount of weight for flexi in either case, you would want to have access along the route to secure it, especially if it is insulated. The aluminized plastic would need more support than aluminum.
Without site inspection, I couldn't say how necessary insulating the duct would be. Using insulated duct would be a good "better safe than sorry" move but will make installation without full access a bit of a bother.
If the masonry is gone and there is just a framed out hole then galvanized would totally work with fairly easy install with the help of a second person or some clever use of support. You wouldn't want to just blow air through the cavity without ducting as it would be horribly inefficient and tank the air velocity across that much distance.
Adding an in-line towards the middle does mean having controls up in the attic more involved and you do want the Mrs to have control over the fan there instead of her having to go downstairs.
If you wanted to circulate the air instead of forcing air with a passive return, then you get into a more complex situation with two ducts and fairly informed placement of intake and return.
Pulling into the attic will help to lower temps and will be the simplest/cheapest option so long as there isn't a wall you can put a window heat pump unit in.
In your case there are a lot of complexities that may make such a simple solution less viable than OPs.
The total volume of the two spaces, the floor and wall construction, if the basement is finished, and the layout of the basement and main floor all come into play.
Adding ducting to the walls may be a major hassle and expense.
If it were my house, I would probably push air up from two spots into the main floor via registers I added or add more returns to the basement and hope the HVAC circulation helps with the top floor without having to add dampers to the main and top floors in both cases.
Depending on the size of the flue, it would be entirely possible to put a 4"-6" stainless corrugated liner in there with a "T-Body" and "snout" going through the side of the chimney in the attic and close off the damper with an opening for airflow. Then you would have in-line duct fans with a rheostat control at both ends pushing air one way or the other. The duct fans that are the same diameter as the duct don't have much pressure, so you would probably want a centrifugal fan to spend once and get desired results. You can buy the fan with the speed controller or they do have ones you plug into. The fans are loud, so the Mrs would appreciate it if it were inside of a baffle box to keep the noise down while she works. How low speed the fans can go is limited, so don't expect to have infinitely varial speeds without spending a lot more.
There also are temperature controllers so you can have the fans cycle to regulate temperature without manual control. You would probably want to put that on only the attic fan if you went that route.
You could use galvanized duct, but it will rot out over time in the flue and it is harder to install vs a stainless corrugated liner.
The fans are $200-400 depending on how much chooch you want. The liner runs $12-16 per foot. The T-Body and snout are about $150-200.
There are bi-directional varial speed duct fans but they carry a premium, typically require custom duct manifold/plenum fabrication, and are two fans put together to achieve the pressure required. Going with one fan at each end is easier, cheaper, and easier to repair should one fan fail.
If it were my project, I would try just having the fan in the attic pulling and control it with a speed control plus temperature controller. If that wasn't enough is when I would add the fan in the basement and do manual control with variable speed and have both fans push. I don't really see why you would want to pull air from the attic, but you seem to feel you would need to.
It can be tricky to get the snout on the T-Body, you will want some 1/4" extensions and impact, as well as someone who can help you.
If you keep one eye closed and expose the other to sunlight, you can see the difference. The lenses tint a dark shade of purple. I have dark brown eyes, so you can't really notice the difference easily. There is a purple ring that is most noticeable outside of the limbal ring. They don't turn your eyes black like you had the tint of sunglasses or transitions glasses, which would be cool.
I would imagine someone with lighter color eyes, like really light blue, would have a very noticeable difference.
Something I did notice as the wearer is when the lenses are tinted there is like a contrast filter on your vision so colors look better.