Bathroom subfloor replacement
1) Nails have much greater shear strength and are better than screws when that is a concern, for your application screws are fine. I would use 1 1/2" but 1 3/4" will work too.

2) No.

3) Screws are fine, long enough to grip tight.

4) 1/8" - 1/4" gap? No problem. If you are going to tile it won't matter at all, if it concerns you cover it with tile backer-board or trowel some tile mortar into the cracks.

5) I use screws because they are easier to seat flush by countersinking the heads. You cannot leave the heads sticking up (screw or nail), even a little, or you may end up with cracked tile.