Yes you can rent a car in Kyrgyzstan. From a reliable company and at quite affordable price. A lot of options to choose from. IronHorseNomads car rentals based in Bishkek, you can book a car in advance and pick it up right in the city or you even can order the car delivery right to the airport for a small additional fee.
If you are going to Aksuu Yurt Lodge you can park a car nearby the pickup point at the Aksuu(Teplokluchenka) village. Later on you can visit some nice gorges all around the Issyk-Kul valley to ski, relax in the hot springs or just look around. We will provide details for any places you like to visit.