touch a new file, I see that its modification time is showing as the current time in my local time-zone rather than in UTC. If I were to copy that file to a machine that's in another time-zone, would it update to that machine's time-zone or would it still show the same time it showed on my machine? Do file modification times contain time-zone information?
They should be in Unix time: Seconds since 1970-01-01T00:00 UTC. If you move it then it will be the same time. However time is displayed in local time.
The above is true on Unix file-systems. On at least some Microsoft file-systems the time is stored in local time (for backward compatibility with MS operating systems).