When it says to spend a gem to draw cards equal to my number of distress tokens, I'm assuming I can do this at any time during my turn... Sooooo...
1) Can I spend my cards, then use this to get more cards and keep going?
.......1a) Can I spend an amount of influence (cards) as intent, use the mask to increase the cards I haven't spent yet (to my hand limit), and then take from those new cards to add to the amount of total intent involved in an intended action?
2) Were I to be at (or near) my hand limit when I use the mask ability, do I have to discard cards in order to draw more (to stay under my limit), or does this mask ability temporarily ignore my max hand size limit during this time?
3) ...Does this mean this ability ceases to be use-able after the 6th distress token?
1) Yes, just keep in mind that this is a "once per turn" ability.
1a) It is not clearly stated in the rulebook which abilities you can use while performing another action. Personally, I only allow abilities/actions that directly give me intent for my current action. So my guess would be no, you can't partially pay for an action, "interrupt" the action to draw more cards and then continue paying for the same action.
2+3) According to the rulebook, you can never have more cards in your hand than your hand size limit (normally, 6 cards). So, no matter what ability/action instructed you to draw cards, you just stop drawing if you reach that limit, you don't draw more and then discard down to 6.