What type of screws would work best for hanging a projector from the ceiling?
I'm assuming your projector isn't terribly heavy, so...

2 1/2" gold construction screws should work fine to hold the board to the joists. I also like to use decking screws because of their strength and color.  They both have small enough heads that shouldn't stick out once they are tightened.  I think that larger fasteners like lag bolts just aren't warranted for this project.

The 2 1/2" screws should end up going into the joist about 1 1/4".  If you use a thicker board for support, you might want to bump up to 3" screws.  Try to get more than an 1" into the joist.

Since this is interior, and not a large span to tremendous weight, pine or MDF should be fine.  MDF will be easier to work with because it will not be warped or twisted.  I used it to mount my TV to the wall.

Any wood screw that goes into the full thickness of the mounting board should be fine to hold up the projector mount. Even if the screws are a little long, they will just poke into the drywall a bit, which will be fine. Use a screw as thick as possible based on the holes in the mount, and use as many screws as you have holes for.