Does it freeze in winter where you live? Ice expands as it forms and this can do damage to masonry. Since your chimney is open to atmosphere, rain water probably evaporates faster than it accumulates. It may be persistently damp in there though, which means mold. Nothing a bleach solution in a spray bottle can’t fix.
It freezes but I guess I wouldn't expect pressure on the masonry because the chimny seems to be a cast iron tube so I would think the expansion would be forced to happen vertically.
Yeah, you may be good to go, structurally. I think most modern chimney liners are ceramic. I struggle with drafting issues, aka getting smoke to go out.