I just went through this:
1) If the framing under the toilet is water saturated and rotting, obviously cut out the floor, and box frame and place new plywood around where the waste pipe and flange would go. You'll have to cut the waste pipe and then use a coupler to reattach the waste. Reinstall a new flange at proper height once the flooring is in, or before if you know the finished floor height. If the wood is not rotting and wet, then just use a flange extender. The one I bought came in a kit with both 3/8 & 5/8 plates that can be used separately or stacked.
2) With the correct height flange only 1 wax ring, without horn, will be necessary.
3) Can't really comment about best toilet, as price for you might be a consideration. Toto Drake gets great reviews around here. I have an AS Cadet 3 that has been reliable and easy to install and reinstall.