Yes, install your sheets 90 degrees to the existing subfloor. Yes, screw AND GLUE as well. Use a chalk line to find your floor joists. If you can at least screw into them THROUGH your existing subfloor, you should be good to go. Just screwing into the subfloor is a little bit of a gamble. Be SURE you let your new sheets get 'acclimated' to the house BEFORE you install them. Otherwise, you may end up with unwanted buckling or gapping. Lay them out flat for a few days, then flip them and let them lay again for a few days. Use a high quality screw that's charactarized as a Finished Screw. DEFINITELY predrill your holes, or you may split your wood. (Since your plywood is a 'finished surface', you want it to look the best!) Make sure your screws are an equal distance apart from each other, again, for 'looks'.
Now, my thoughts on your project ... It will certainly LOOK real nice. Granted, the bedroom floor doesn't get a whole lot of wear-and-tear but it will wear. Remember, you're WALKING on a painted/finished wood surface. At the same time, you've installed a finished floor that now requires maintenance. I'm not sure how long it will last but you may be back at refinishing again in a few years. If the room is real bright, you'll certainly notice the wear that much more. These are just my thoughts and some things you should consider.
I will add that when we bought our home, the Sellers didn't finish the kitchen floor and temporarily painted and stenciled around the borders. It did look nice but it never looked like a finished floor. It looked like a plywood floor that was painted and stenciled. You'll want to use the BEST plywood you can find. Your floor should be sanded BEFORE you start finishing it. It should be primed too. Use the BEST material you can pay for. (You're only finishing one room so the extra $$$ you pay for high-quality paints and such will be well worth it!) Visit your LOCAL Paint Retailer (found in the YELLOW PAGES under PAINT - RETAIL) for material AND tools.