By "drywall screws", I think you are meaning 1 5/8" blued screws. I do not think that they would be a good choice. We usually shoot down plywood subflooring with 2 1/2" to 3" ring shank nails, however 2" to 2 1/2" decking screws work well, but just a lot more work. I would encourage you to use a little adhesive on top of the floor joists or strips of felt paper or Grace ice/water shield to avoid any squeaks later on.
The plywood goes down so the groove is away from the wall, that way, after the row is finished, or at least when you decide to start the next row, a 2X4 can be placed across the joists at the edge of the next row and drive, with a sledgehammer, the second run into the first. The gap should be 1/8" between ends. The tongue and groove (T&G) of the sheets should be driven snug. Do not try to drive them tight, although in some cases it might be needed in order to keep a straight run. The first run is set to a chalk line, but the rows after that can start to meander after a few runs are down, and to correct it with a tighter fit or if need be, a loose fit to correct the run so it remains true
The subfloor adhesive I choose to use is polyurethane based adhesive. It has a long open time, Meaning you don't have to rush to get it covered before the glue "skins" over, thus minimizing the bond. Also, it is twice the cost of regular subfloor adhesive, but you can actually use less. Since there is no "skin" that will form that will need to be broke again by the laying of the plywood over it, to squeeze the glue again out of the skin that was formed by it sitting open before the plywood goes down. Do not drop the plywood on the glue or set the plywood off the final placement too far or firstly the glue will splatter all out off of the joist, or slide or scrape the glue off the joists. I seen both of these things happen in my years of construction. Needless to say the bond will not be worthy. The tongue or the groove does not need to be glued, although there is no reason not to. Some engineers require it in certain applications, but for standard subfloor applications none will be needed. Again it is to your discretion also. If you choose to place the glue at the groove, so the tongue drives it into the groove.
Screws are the best to secure the plywood with, and since a premium glue is used, there only needs to be enough set to draw the plywood to the joists. You can still place them as it has been done before the days of glue which is 6" on the edges and 8" in the field.
With glue you really can spread them out farther, 5 screws used on the ends if you choose, and 4 across the field. There are drills that are sold that use screws that have the screws collated in coils or strips. Makes this go much faster. Rental is an idea here.
Again with the glue, nails can be used too, since the glue will do all of the holding after it has set. Air powered nailers are available for this and will be faster than screws, but the nail gun in some cases does not draw the wood together like screws.
Best Answer
If there is another space below, then you want the subfloor for no other reason that to prevent dust, air, sound and light from leaking between the floors.