OmniMount OCM On-Wall Cable Management Covers, Paintable, Flat Design to Conceal up to 6 Cables, Set of Three, 3.25" x 18" Inches

Q: What is the depth of this off the walls ? How thick is it?

The depth of this unit off the walls are possibly around 1 inch thick.

Q: Do you need to screw into a wood stud or can you screw it only into Sheetrock?

Both but we ended up with screwing into a wood stud.

Q: Does this allow for the cables to slide through it easily as you pull the tv out from the wall and push it back later?

That will depend on a number of factors, including the amount of slack in your cables and the thickness of the cables. The cables are held in place by clips on the inside of the cover, so if your cables have sufficient slack and aren't too thick or numerous, they should be able to slide as you move your TV.