It sounds like you have several issues: A) Squeeks B) Cold Floors C) Concern about floor level differences Regarding A, the builder is basically correct - 3/4 OSB is a building standard now. It is tongue and groove and glued down and together. Over time is will begin to squeek. You know where the squeeks are and when removing existing flooring, nail the subfloor tight again. Should eliminate, or at least reduce the squeek. Regarding B, See if you can have foam insulation injected into the cantilevered space. As to heating, it still may be necessary but why not minimize its burden by controlling the root cause. Regarding C, Several things here. Your current vinyl should be installed on a thin underlayment. Combined they are about 1/4" thick. It will be necessary to remove these. (This will be the time to inject foam insulation) A tile floor needs 1.25" minimum structural support which means a 1/2" thickness of concrete board will be cemented and screwed atop the existing subfloor. Then your heating mat followed by tile. So, you'll be gaining about 3/4" above the current level. Your tile shop will have a marble threshold with a beveled edge to minimize this effect. We have this same situation is our home and it is negligable. You may or may not have to trim the door. Base trim would be removed in the bath to do this work. It can then be reinstalled, replaced or have the tile setter install tile as base trim - that's what we did. Here is a link to an updated bath we did and amongst the photos you can see the threshold and tile base trim. Hope this helps. Good luck. Here is a link that might be useful: Bathroom

My last tile job was off white ceramic. We put a slab of the same color granite at the same level as the as the tile and it looked great.

Is there a height difference? I see two options. It looks like you have extra tile in the pictures, if so you could cut the wood to an even margin and fill the space with cut tiles. Another option is to cut the wood floor to an even margin and then fill in the gap with wood running in the other direction in a contrasting or color matched stain. You could run the new wood in the direction of the existing floo but all those short pieces in a row would look odd to me.

When you say tile, do you mean permanent flooring (porcelain tile, ceramic tile, stone tile, glue down vinyl tiles, glue down cork tiles, glue down linoleum tiles)? Or do you mean a floating floor 'tile'?