patching hole to be reused
If you only drilled a 1/2" into the stud, and you know the stud is deep enough, get a lag or screw that matches the others a extra 1/2" longer. No doweling needed. You can dowel it if you like, but a dowel, a 1/2" in diameter and a 1/2" in, really does not hold much, the screw will still need to be 1/2" longer. 2"-2 1/2" screws should be the norm, so a 3" screw will work, unless your screws that came with the mount are 3 1/2", then the 4" screw you suggest will be fine.

Enough is perfect, too long can be bad. An 1 1/2" into the framing is great, in your case it will then need 2" not including the wall finish. That is never considered part of the length that the screw goes into to hold items in place. If your wall material is 1/2" thick, and the metal is thin, you can use a 2 1/2" screw if your wall material is 1" thick, you need a 3" screw, and so on.

Edit 3/16/15

I added a sketch to show what I am referring to. I hope it helps explain.