Yes, but first remove the tile...:-)
Actually, the trick is to remove the tile without gouging the hardwood floor beneath. Some surface scratching is inevitable while removing the tile but you want to keep it to a minimum. The sanding process when refinishing will eliminate any blemishes that are not to deep.
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That depends upon location. Tiles can be installed on a gypsum board wall or ceiling. (But make sure that the tiles are well grouted; if moisure gets past the tile into the gypsum board, it will be damaged.) Floors require greater strength - but then, you wouldn't normally have a gypsum board floor, with or without tile.
This is not a good idea without using a cement /mortar board underlayment.
Not very thick. Many building codes (perhaps most) allow 3/4 inch waferboard or 5'8 inch plywood as the structural subfloor over 16" O.C. joists. This is fine for normal sized floors with carpet or vinyl overlay, but is insufficient for very large floors or floors to be finished in ceramic, slate or porcelain tiles.
Tiles used on floors tend to be thicker than wall tiles. Also, floor tiles are subjected to flexing as people walk over them. Therefore, if floor tiling a wooden floor, use a flexible cement adhesive and grout designed to give and take without cracking.
Try not to invite Aretha Franklin and Queen Latifah over for dinner and your floors should hold up just fine.