Are Shimano 105 brifters (5800) compatible with Avid BB5 mechanical disc brakes?
Yes they are. I had Avid BB5 calipers with Ultegra 6800 levers (same cable pull as 105 5800) on a Specialized Tricross and they worked very well together.
All road brifters use a similar cable pull ratio for the brake and you'll need a matching set of brake calipers. There are two different BB5 models, one for MTB pull ratio and one for road. It will work provided you get the road version.