Yes they throttle themselves, and will shut down before burning a hole in your bag. If it's shutting down due to overheating it's possibly a bad thermal sensor. I've usually only seen this on systems that have had Liquid damage or a device that was previously repaired.

M1 macbook air owner here, it absolutely heats up when working hard, especially at the top of the keyboard.

It doesn't catch fire or anything, but it's hot enough to be unpleasant to touch, I don't doubt it's hot enough to cause some expansion in the screen (especially if closed).

It's not really "issues", but under heavy workloads the SoC will absolutely stay above 90 °C for extended periods of time, which, especially in clamshell mode, can cause the bottom part of the screen to heat up a lot.