In Tbilisi, my hotel of choice is Betsy
http://www.betsyshotel.com/The atmosphere is friendly and cosy. Being located over a hill, it has a great view over the city. The walk to the city centre is steep but takes only 20 minutes (5 by taxi).
Some rooms have nice paintings hanging on the wall (see picture below - you need to register to view it).
Dinner is included in the room rate. You often share the table with other guests.
Friday evening is TGIF at the lobby bar and expats meet there.