I want to make sure something I did was legal, because it seems like it shouldn't be, but I think it is.

Unit A moves adjacent to a enemy unit and attacks at half strength close range and becomes fatigued.

Then Unit B moves adjacent to the same enemy unit and assaults it with support from Unit A. (which is fatigued and has already done a Fire and Move)

Then Unit C moves adjacent to the same enemy unit and assaults it with support from Unit A and Unit B. (which are both fatigued and have both already attacked).

Is the turn described above all legal?

yup - all legal. Welcome to the wonders of the assault wave. The most devastating and effective attack in ToI. :-)

Notice that that flamethrower Squad A had now has been in effect three times!

Alson notice that it could be a diffrent defending hex.

All perfectly legal.