If you plan on storing some heavily-weighted items up there like the bottom half of a car engine or a few cinder blocks or the like, use 3/4" there. If you're storing general 'stuff' up there, then the 5/8" should be fine. As Henry says, just keep in mind that you're spanning 2' and if you weigh 130 lbs and your 5/8" sags, then be wary that over time, the sagging can pull up the nails that hold the board down.
Certainly, you can put 2 x 4's between the joists. If you do, use the appropriate 'hanger' for a 2x4, and put them no more than 16" OC. Wherever you put in, and cover, 2x4's, I'd say you're pretty safe with using 5/8" sheets. My best to ya and hope this helps. (Henry, please add-to / correct this if you're up to it.)