pressed in faucet seat evaluation
Yes. Maybe. Yes.
To remove a seat use this tool 
You can see at the 10 Oclock position on the seat (2) chips are missing which may cause a opening when the washer is pressed down to stem the water flow.
A beveled washer may make a complete seal rather than a flat washer, but your better off, as you mentioned, to replace or grind the seat. Check/replace both. Replacing is probably the easiest and most assured  way to repair the faucet. Any well stocked hardware store will have an assortment of seats. Take the damaged one with you.
A seat dresser will have a guide for the dressing wheel that keeps it steady and at a 90 degree position for a uniform removal of seat material.  Replacing the seat is easier and more likely to be successful at repairing the problem.
stepped washer