Here in Oz we have 3 basic types of ply,, cheap, structural and marine. I presume its the same in the states.. They come in different thicknesses too. The cheap ply buckles with extreme temperatures, sunlight and moisture. Structural is way more resiliant against this and marine grade ply supposedly weather resistant is the best. Just remember the thinner and cheaper, the more uneven it becomes later.
Normally timber gets glued, screwed and or nailed but in your case you need to keep to the tape idea. There are some removable tapes now like scotch 3m.
Using 2mm foam underlay will protect your existing floor the best but gives it that bouncy floating feel we use under floating floor systems.
To fasten each sheet together with no movement or damage to either floor, try buying t floor dividers, (look it up on google images), cut them to short pieces and countersink them into the ply by using a router or similar to recess the ply, and screw them using screws shorter than the ply thickness. They come in a range of materials and shapes but aim for flat alloy ones.