Am I missing some way to save in the middle of a level?
Yup. It's kinda unintuitive, specially with touch controls, but you can save any time by hitting pause, going to the passport on the menu, and cllcking "save game" when it pops up.
My issue was that it seemed like some weird glitch (the switch was switched! it showed the door open! and then it was closed!), and then I went up to check and died in the process and have to do the whole of Level 2 over again. Sigh
Sounds like bog standard Tomb Raider. It's been awhile, and can't remember much of the second level, but it's a pretty safe bet to say that it probably closed on you at some point. Timed doors, gates, traps, and what have you are very, very common throughout the game.
It is kind of funny how part of the puzzle and challenge of the game is due to the cumbersome controls, and yet the control style still works as a style choice for the game rather than working against it (once you get used to the way they work).
One thing you say about the controls is that they're a little stiff and mechanical, but you do get used to them. When I first played it back in the day, I barely felt like I could control it when I first started, but ended up being pretty agile and graceful by the end of it.
If you're comfortable with the touch controls, you will (well...maybe) get used to it after awhile.