subfloor
I removed the tile in my bathroom. The top photo shows what it looks like below the removed tile. The bottom photo shows a closeup profile, near the toilet flange. It is 1 inch think with wood plywood at the bottom. The checkboard pattern seems like paper when I peel it. Tt is is very hard and solid and only chipped away in some spots over the entire area.
My questions are: 1. Can I tile over this? 2. What would be the surface prep.? 
Thank you
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.
1/2" plywood is for temporary applications, exterior sheathing and used as gigantic shims. It does not belong as a floor or a roof; minimum 5/8" exterior grade.
What subfloor should HardieBacker board be installed over? The minimum subfloor specified is 5/8" exterior grade plywood or 23/32'' exterior grade OSB subflooring. HardieBacker board must be installed in accordance with local building codes and the floor must be engineered not to exceed the L/360 deflection criteria, including live and dead design loads, for the specific joist spacing used. HardieBacker board is not designed for use over concrete.
Subfloor should be minimum 5/8" exterior-grade OSB or plywood.
For the subfloor, securely glue and fasten minimum 5/8" (16 mm) exterior grade plywood or OSB panels (PRP-108) to the floor joists. Floor joists should be spaced a maximum of 16" (40 cm) o.c.; a 3/4" (19 mm) Exterior Grade Plywood or OSB subfloor with I-joists spaced a maximum of 19.2" (48 cm) o.c. is also acceptable.
For the threshold you don't want something that overlaps, like this:
For a Hardwood transition, you want the type that abuts the tile, like this one-sided beveled threshold: (I've had luck filling against them with grout; complementary colored caulk is also available for tile-to-wood transitions. Standard thresholds can be ripped down, length-wise, to remove one bevel and then the bottom sanded down for a custom fit. Endeavor for absolutely perfect cuts along the doorway.)
Since you are transitioning to Carpet, I'd use something like this or this: (an abrupt leading edge is more of a concern then a reasonably high threshold)
Best Answer
It looks like you have backer board looking at the edge of the toilet hole. Since the pattern feels like paper it was probably a mastic put on the tile or back buttered as Huesmann stated but We normally put adhesive on the bottom surface with a notched trowel.
I would attempt to peal / scrape this material up since it has come up in areas your new tile may not set as well.
If the material is paper thin you mentioned felt like paper and you are using large tiles you probably can tile over this without it showing. If it is a mastic many are water based but make one heck of a mess if it gets wet , it easily plugs sanding belts so if it won’t scrape up clean the area and overlay with your adhesive or thinset on the surface with a notched trowel.
Back buttering is coating the tile with the adhesive or thin set then scraping most of it off it wets the surface and you get a much better bond.