1. Dos-Box crashes everytime I press the esc key, to skip to the main menu. This did not use to happen.

2. I can't get a pad to work. When I'm in the main menu, the options scroll non-stop and I can't select anything, even after I remove the controller. This is with a DS4 and Xbox 360 pad.

Try the native Windows version:

https://www.gog.com/forum/tomb_raider_series/tomb_raider_1_native_windows_and_resolution_patch

See my reply in that thread for instructions and links.

It's a bit of work to set up, but I suspect that it'll solve both issues.