Whenever that happens, it's just that your HUD didn't reappear(for whatever reason; it's a somewhat common occurrence). Press Shift-O, and you can continue normally. I don't know/recall if the friendly mecha does anything.
The first mecha's weak point:
Look for the Shivan Comm Node, near its chest, IIRC. I recommend bringing Maxims to kill it from a distance.
In the last mission:
I've actually forgotten what 'objects' were used, but...inside the Hades fighterbay, at the 'groin' of the mecha, are a pair of objects, probably cargo containers or something; you can't fly inside, but you can shoot them from outside. Destroying both will cause the mecha to self-destruct.