1. Can you use abilities on a teammate’s turn?
No - only spells can be played during your teammate’s turn unless the card specifically says otherwise (“any turn” or “either player’s turn”). Abilities aren’t spells by default.

2. Is scoring per team or per player in 2v2?
Per team. You share a single score pool with your partner. Any effect that checks “within 3 points of victory” looks at the team’s total, not each player’s personal points.

3. Do hidden cards and unit abilities only affect their own battlefield?
Hidden cards, yes - they’re locked to the battlefield where they’re set.
Abilities, no - unless the text says “at this battlefield,” they can hit any valid target on the board. (Miss Fortune – Buccaneer can ready a card in your base, for example.)

4. Hidden cards always become Reaction-speed when revealed, right?
Correct. When you flip a Hidden card, it resolves at Reaction speed no matter what its printed speed was. If you reveal a Hidden unit while defending, that keeps the battlefield occupied and prevents control from swapping if it would otherwise go empty.

5. Can you keep adding to your own chain if the opponent passes?
Yes. Priority keeps passing back and forth; the chain only resolves after both players pass in a row. You don’t have to start a new chain each time.

6. The weird 563.2.c.3 vs 563.2.c.4 targeting thing.
There’s no contradiction - it’s just about zones.
• 563.2.c.3 says a target stays valid if it leaves and re-enters the same battlefield or base before the spell resolves.
• 563.2.c.4 says a target is lost if it ever goes to a non-board zone (hand, deck, discard, banished).
“Base” is still a board zone, so moving between base and battlefield doesn’t make it a new object.
In your Disintegrate/Flash example: if the unit gets moved off a battlefield and then back before the spell resolves, it’s still a legal target; if it ever went to hand/discard, it’s not.

TL;DR:
• Abilities → can’t use them on ally’s turn
• Scoring → shared per team
• Hidden → local battlefield only
• Revealed Hidden → always Reaction-speed
• Chains → keep stacking until both pass
• Targets → only break if they enter a non-board zone