Say for some reason, a contested area (containing units from both side) does not contain an activation/route counter of mine. It is still my initiative to act. Can I issue an attack/move order that originates from said contested area?

Yes. Also, if you should attack the contested area from another area, your units in the contested area will contribute to the battle.

TK

Ditto. Just like TK explained!!!

Great way to consolidate forces and get scattered singleton units out of harms way: play an assault and pull in your contested units (assuming you win) and form up a single powerful force.