Tile can go over tile IF the bottom layer is clean and well bonded to the sub floor. If the layer to be covered is at all loose, both layers will come up. Remove any loose grout. Remember: stacking in this way will add to the thickness of the floor, which can be an issue, depending on what the neighboring floors are like. As to grout lines: there will be spaces between the tiles, it isn't possible to eliminate them completely. You can either fill them with nice clean grout, or they'll fill themselves [courtesy of gravity] with bits of dandruff, body hair, spilled milk, sweat, etc. Personally, I vote for grout. How wide the grout lines ought to be depends on the tiles you are using. Most tile is molded to size, then fired. The firing process can cause the tile to deform or shrink slightly, no two will change in quite the same way. The grout allows you to compensate for this. For 12x12 tile, you'll generally want at least an eighth inch grout line, larger tiles generally have larger lines. But some tile is 'rectified', which means it goes through a saw after firing. These are closer to perfect in dimension, and so can use a narrower line: 1/16th usually works for 12x12 rectified. Going narrower may be possible, but requires a higher level of skill than most first timers will manage. You can usually spot a rectified tile by looking at the back and edges. The saw may leave marks, often time it passes through the mold marks in an obvious way. If you aren't sure, contact your tile seller, they can tell you. They should also be able to advise on grout lines, but be wary if they claim you don't need to grout at all: sure sign of someone who doesn't know what they are talking about. Sanded grout is stronger: the sand works like bricks in a wall, and also resist abrasion. Therefore preferable in most contexts. Traditionally, sanded is used in grout lines of 1/8th inch or larger, unsanded in spaces less than an eighth. This is because the sand matrix doesn't like to be squeezed into really tight areas. Recently I've noticed some sanded grouts that claim they'll work in 1/16th lines. The salseman said they are using a finer ground sand to make this possible. Sounds reasonable to me... You may want to check out the John Bridge forum for more detailed info, or ask on GW's 'Bathroom' board-- that's where our resident professional tile gurus [I'm only an interested amatuer] hang out.

Tiling over tile in dry areas(not bathrooms/showers) is one thing. I agree tiling over shower tile is a bad idea. Even if the existing tile is in good condition---and(huge and) the waterproofing(if any) under the existing tile is in good shape, adding another layer of area where trapping water is a probability is simply creating a future problem.

Not a pro (and I’d hire one to do this!) but ripping out and starting over will produce a better result. The tile on the floor should be level to the wood. Looks like it’s lower now. Also adding metal trim schluter to all exposed edges would finish it off. Also Consider pulling the interior wood trim (which they tried to cut around) off and replacing it post install.

Having trouble getting photos to attach. Trying again...