Seems like the pyrofuse probably blew.
No idea on the total (I'd guess ~$10-$15k of damage, but it depends on how many pieces they have to replace...very hard to know), but remember if it is repaired to claim diminished value (you make this claim after the car is returned to you completely repaired). Not sure exactly what your car is (year model), but the diminished value will likely be over $3k. There are calculators online to figure it out - typically diminished value amounts cap out at around $10k (beyond that the car is likely a total loss). So it'll probably be in the range of $3k to $6k. It's calculated by the amount of damage to the vehicle, etc.
You can only claim this if you are 100% not at fault (the apparent situation here, since you didn't hit anyone else after getting hit).
Again, after the car is 100% repaired, you do this. There's no need to rush it or even mention it earlier (the claim is not closed after your car is repaired). You just write a nice letter saying you have to disclose the accident to future buyers, etc., etc. Common sense stuff, and the insured's company will just give it to you, though of course they will try to negotiate (otherwise you take their insured to court, which is not a good look for them).
Of course, this will only work if the person who hit you has a high enough property damage limit. (In California the minimum allowed is $5k!!!). So hopefully you're good there (talk to the insured's company).
In general, for the damage, whether you go through your insurance or the insured's is up to you - my insurance company has a "interim" state where you can get counsel from them about the best course of action prior to actually opening a claim. It can depend on what you have on your policy, the at-fault insured's policy, rental reimbursement rates and limits, etc. There are a lot of factors which differ in every case, but in the case of not being at fault it can be nice to go through the insured's because you won't be out-of-pocket for your deductible (you'll get it back through your insurance through subrogation (since you're not at fault) eventually, but you'll be temporarily out of pocket). Anyway, talk to your insurance company.