Is it permissible to perform a single aqiqah with the intention for both a newborn and a car, and is it valid to do so before the full price of the car has been paid?

Aqiqah (definition) is a recommended Sunnah (definition), not obligatory. If Aqiqah and debt repayment conflict, debt repayment takes precedence. The intention for Aqiqah is required before slaughtering. It is permissible to combine the intention for Aqiqah with the intention for a celebratory meal for an occasion, such as buying a car or averting misfortune, because the celebratory meal is implicitly achieved. Combining the intention for something that does not require an intention, like a vow or a celebratory meal, with the intention for an act of worship like Aqiqah, does not invalidate it.