New Drywall poorly installed
Hopefully we can get you back on track


  he put the drywall flush to the floor. I know it needs to be off the ground 3/8 inch or so


I tend to hang drywall from the top down for this reason. It sounds like he might have started from the bottom up and left them sitting on the floor. You're right about needing a gap from the floor (I use a super bar myself to lift up when mounting)


  Can the wall be trimmed in place or do I need to take it down first?


You can trim it in place. I would suggest buying some chalk string and snapping a line around the room, then take a box cutter and cut along the line. You could also use a drywall saw, it's just harder to cut around the studs.


  He has only put 2 screws per stud and very few of them are at the floor level.


I assume we're talking 4' x 8' sheets. Two per stud is really slim. I would want a fastener (nail or screw) every 6 - 8". Two per stud can hold it up, but you don't want the board to move and that's not enough to keep it from doing that.


  all the sheets are flush together, whereas I thought there should be an 1/8 inch gap between sheets. Is there any way to fix this or should I just hang new sheets with the gap?


Nope. In this case your installer is right. The sheets should butt up to one another. You can leave a gap, but that's just more mud you'll need to apply.


  some of the screws are sunk in quite deep


Sounds like he's driving screws using a regular drill instead of a drywall screw gun (which is designed to sink them to the correct depth and no further). You want your screws to sink into the surface some, but just enough for you to fill the hole in later. If they've gone most of the way through the board then they're useless. 

If you don't feel confident enough driving screws yourself (it's a bit of an art without screw gun) then use drywall nails instead. They're faster and cheaper. Just bash the heads in enough to make a dimple in the drywall (makes skimming over them easy).


  Can I add screws and cover errors with tape and mud?


Regardless of whatever fastener you use, you should mud all the holes. That's more or less what mud is for. Even if you make a mistake, mud covers all sins in this case.