Let's see where to start:
Part of the difficulty of making a team game is giving players a chance to strategically place less strong players in less important positions without taking them out of the game. Family Feud was great at that. Your concept really fails there. One player from each team doesn't get involved until the third round. You're going to have to find a way to get all of the players involved more. For example, let each of the six players choose a category in round one, and give less time per category.
Opponent's choice, again, eliminates one of the players in each round. "You're the weakest player, take a powder." Again not good for team spirit, or for airing. Instead, try this this variant: The captain from each team plays solo. The two non-captains on the opposing team together choose a subject. In addition, they get to try to steal any questions missed by the captain. This adds extra strategy to the category selection. Do they try to choose an opponent's weak category, or one of their own strong ones? Kind of a variant of a common quiz bowl round.
Your writing about round three may be unclear. The way you've spelled it out is that whoever's behind tries to play catch-up at $100 / quesetion. What does the other team do? Additionally, bidding to answer against the clock add an extra variable that make things unfair: the length of the questions. One category could be artificially easy or hard just because the questions take longer to ask. You might consider borrowing from Debt, and make it a set of five questions (or 10, or whatever) so that question length is no longer a factor. Besides, it will help to have a different cadence in a round, instead of everything being against the clock.
The set-up for your bonus round does intrigue me, though. It's not simply who knows the most about their area, it's whether your own partners are comfortable in your own area. Nice twist, IMO. It won't seem as original to some given what ESPN Trivial Pursuit used earlier in the year, but I still like it. Obviously, it's hard to do this in a returning champion situation without making new categories each time, so you seem to have thought of that.
You're going to get plenty of comments about the way you've set up your budget (and they'll likely be valid). What do the other two players in that round do that merits them a car as well. That reeks of mo' money. Instead just give them the money. Or to see some real fighting during category selection, dump the $10k and let them each choose from among three cars in the order they played the final round. But don't give both.
EDIT: scratch those last two sentences. That idea penalizes stronger players. Just give them the money and call it done.
Finally: Where does the isolation booth really come into play? The booth as a gimmick is only useful when there is information you need to keep from a player (the other person's score in 21, identical questions in WBSM). Here, there's no reason to have a booth. Without that, there's no real gimmick except that it's a rapid-fire quiz. This community in general is partial to that type of format, but it's not nearly enough for the general public.
So in addition to tweaking the gameplay to allow more participation by all of the players, and the budgetary concerns, you need to find a gimmick that actually affects the game.
