Assuming your new flooring is a floating system and the existing tile is flush with the existing subfloor, there's no reason why you can't float over the tile, too.
The flatter your prep work is, the better your finished product will be. Check with the flooring mfr. on tolerances allowed and on what leveling products (if needed) they recommend.
There are cement-based and gypsum-based leveling compounds.
Stay away from Fix-All.


Yes, but again, check on allowable tolerances from the flooring mfr.


Without knowing what your new product thickness is, I would say to check with the mfr. to see what reducer profiles he offers.
If you're not good with their reducer offerings, and can't find a substitute elsewhere (fab yourself) then the tile would have to demo out and the subloor built up to the correct height for a nice "no bump" transition.