The shortest trip you can take from Tbilisi: just 30 minutes. A small town that was Georgia's capital 1,500 years ago, with a hilltop monastery above the meeting of two rivers of different colours.
All year — half a day is enough in any season.
There is no train and no comfortable public transport between Georgian cities. The practical option for a family is a car with a driver who meets you at the airport, stays with you throughout, and knows the mountain roads.
Yes, especially since it does not cost you a whole day. It is usually added to arrival day or to the Kazbegi road.