OMG. this is not rocket science.
OP.. if the other party has been proven to be at fault by a police filed accident report, the other party is liable for all damages. This includes your car, anyon elses property involved and any medical claims.
They were insured correct??
If so the insurance company will likely be contacting you very soon. If not, you need to contact them. If you do not have any success on that front or are not happy with the compensation package they have offered you, ask your insurance agent to get involved. In most cases thoush it will take an attorney but at the end of the day, they will have to pay that expence for you as well.
Absurd to think you could be jeopradising yourself by trying to call and ask them for a rental car. this should have already have been offered.