Hello,
In answer to your questions
1) Yes, But, if you travel to the very rural regions of Georgia such as Tusheti and/or Svaneti, this could be a problem. In Tbilisi, the Kakheti region etc, definitely no problem. If you are going to August make very sure the hotel has good AC, as the temp can be 40C. Hotels Kopola, GTM Kappan & Old Metekhi can be good choices.
2)Yes, usually. In Tbilisi I would suggest seeking private care at Medi Club 22a Tasknent or CITO group. Good family travel insurance with medical evacuation benefits is important. The Travelex Insurance is more likely to be accepted. Call your Embassy for more info. CITO or Dentaplus are good dental choices.
3 Hiking is everywhere. Buy a good travel guide from Lonely Planet or Bradt for details. The Kazbegi region offers numerous hiking opportunities. The rural regions I mentioned earlier are somewhat questionable for young children. These areas are perhaps best done with a guide/driver.
4) Self drive is easier than you now imagine. However I would suggest renting your vehicle at the Tbilisi Airport to avoid Tbilisi traffic, which believe me you want to do. Kakheti is an easy drive and I think Kazbegi would be OK but the 22 km of the pass is interesting, dirt surface with pot holes. If you drive to Kazbegi I would say book ahead as August is quite busy but also cooler at its' altitude. Of course, all of the above will be greatly simplified, exceot for the money part, by hiring a car with a driver. You could always drive some yourself and hire a driver for the questionable routes
5) Yes, but that depends on your desired route. Kazbegi is 4 hours, Kakheti 2, but more rural areas can be a bumpy all day adventure. I would say avoid mini-bus transport or shared taxis, unless you want a hot crowded vehicle, rather high speed ride, usually with many young people holding their backpacks on their laps.
Best wishes for good family travel Erez Ralph