Veritas Standard Wheel Marking Gauge

Yes, it's built primarily for across-the-grain marking, but it also works along the grain or through end grain, though a knife edge like this one can follow the grain in coarse woods such as oak if you're not careful.

Either works, since the cutter marks in both directions; many woodworkers get better control pulling the gauge toward themselves, but pushing it along the wood is the more traditional technique.

Loosen the thumbscrew, then slide the rod through the body until the wheel of the standard wheel marking gauge projects the distance you need, using the internal O-ring to hold light friction as you fine-tune, then tighten the thumbscrew to lock it.

Yes, though it's hardened for a long working edge, you can resharpen it by removing the securing screw and wheel, then lapping the face on a sharpening stone.

A wheel cutter, like a knife edge, slices wood fibers instead of tearing them, which leaves a cleaner line across the grain than a pin typically produces, though a pin can be less prone to following the grain along the length of a board.