Negative balances will be reported as $0. It will only "hurt" if you have no revolving credit cards reporting a positive balance on your credit reports, but utilization is temporary and as soon as any account reports a positive balance any points lost would be regained.
As @K-in-Boston says, it will report as a $0 balance and only have a negative impact on your scores if you have no other bankcards currently reporting a balance. But even if that's the case, I don't think it's necessary to go out and quickly spend the $93. Any points you lose will come back as soon as any of your other cards report a balance, and unless you have an app coming up before then where the difference in points would cost you more than $93 in interest there isn't any real reason to worry about them in the meantime.
Yes, you can use any card that reports prior to your next credit score pull for the mortgage. As for timeframe on that particular card, you would just need to use the card and allow time for pending charges to finalize and come above $0 prior to the statement closing date.
The $93 negative balance won't expire. At some point Capital One will probably send you a check for whatever of it you haven't used, but then you can just cash the check. Any of your other bankcards will work for avoiding the AZ penalty (don't use a retail card or true chargecard, if you have any), so you can report your $10 balance on any one of those. If you have a Discover card, they will do off-cycle reporting, which would let you get a balance reported in probably a handful of days.
