The rich and powerful take what they want. We steal it back for you.
1. Can a Brilliant Strokes card that has been used to Trump another card be Trumped itself? For example, if the British use their Brilliant Stroke to trump the Patriots' Brilliant Stroke, could the French then use their Brilliant Stroke to trump the British card?
Yes, as long as everyone is eligible to use their own cards.
2. In what order are Trump decisions made? Let's again say the Patriots play Brilliant Stroke. Can the Indians wait to see if the Brits or the French want to trump the card before playing their own card? I can imagine the decision order might go Patriots-British-French(Brilliant Stroke)-Indians-French(Treaty of Alliance) based on the increasing power of these cards to trump the cards of other factions, but I'm really not sure how it's all supposed to work.
It's whomever wants to trump (and is able to) can trump, as long as the current 1st Eligible hasn't taken an action. Generally, you give others a reasonable amount of time to decide, but that's up to you.
3. Could the French use Treaty of Alliance to trump a Brilliant Stroke card that was itself used to trump France's earlier play of Brilliant Stroke?
Yes. ToA has priority over all other BSs.