cards can still support without supply correct?
Yes that's correct.
and in the event of a tie, are all cards counted? including paratroopers, cards without supply, etc?
Yes all cards are counted irrespective of supply.
also, after all airstrikes are used, how many cards should you start with each hand?
You should always start each turn with 7 cards including airstrikes (the exception is the player taking the very first turn who starts with 6 cards as they only draw and play 1 card that turn).
So if you still have both airstrikes you should have 5 other cards, if you have 1 airstrike you should have 6 other cards and if no remaining airstrikes, 7 other cards.