Touching on the second question, since the ship would never actually reach the speed of light, the trip would not seem instantaneous to the people on board. However, the trip would seem much shorter to the people on board than it would to external observers. The people on board the ship would experience length contraction in the direction of travel making their destination closer to themselves, while external observers would notice the people onboard the ship moving slowly, ie, experiencing time at a reduced rate. Either way, the effect is that the people on board perceive the trip to be much shorter (in terms of both distance and time) than an external observer watching their ship. In principle you can get the perceived length of the trip (both distance and time) to approach but not equal zero, though in practice this would involve killing everyone on board and destroying the ship (and maybe even the galaxy).
I agree with the other commenters that the people on board will experience a consistent acceleration of 9.81 m/s² in your described scenario. It might help, conceptually, to imagine an external observer watching someone on the ship jumping up and down at this near-light speed, taking into account the severe time dilation they’d be experiencing: The difference in perception comes because, from the external observer’s point of view, the person on the ship is moving in extreme slow motion.