Being a pool table owner myself, and a kitchen counter owner. I can easily say I have no idea specifically what type of slate pool tables are made from. I do know they come out of one quarry in Italy. Hard enough for a counter? Yep I'd say so, knowing how hard those phenolic pool balls get knocked around. Normally the felt isn't glued down. It should be stretched into shape and stapled from underneath to some wood that has been epoxied onto the slate or clamped through the slate when the side rails are put on. If the felt was glued on, normally a very light coating of rubber cement is applied, sometimes contact adhesive. The layer is designed to be thin enough that the felt can be pulled off with ease.
Most pool tables have 1" slate. The really really good ones have a solid piece, but that table would be worth more as a pool table than as a counter top. Most pool tables have 3 removable pieces of slate put together to make 1 table.
FWIW its not felt, its worsted wool. Felt is just easier to say.