When is it appropriate to use a university seal as a graduate student, post-doc or research scientist?
It is my personal understanding as a professor and former chair of a program that you can use the letterhead for job searches as long as you retain a position of some kind at the university: postdoc, doctoral candidate, visiting professor, lecturer, research scientist, etc.  

As an appropriate proxy, use the letterhead as long as you have a university e-mail address (that isn't an alumni one).

Once you're no longer affiliated with the university because you graduated or separated, then you should not use the letterhead.

[Note that use of letterhead for job searches is a special case. Do not use the letterhead for other personal purposes - such as writing a letter to your congressperson - if you're in a liminal position as the possible blowback can be hazardous to your career].