Yes, you can but you need to staple a mesh to it. Then fill the mesh with thin-set and let it dry before putting your tile down with thin-set.
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It is possible but I would not reccomend it. If possible remove the old tile before laying the new tile. If you want to avoid removing the old tile you can lay a cement board over the old tile before installing the new tile.
The first thing you want to do is be sure that the original tile is secure and even. If it is loose, or uneven the new tile may pop out or crack under normal wear and tear. If it is uneven or unsecure you can lay a sub-floor to give the new tile a even base.
You should not tile on top of a hardwood floor. If you do, it will crack as the wood underneath will expand and contract throughout the seasons. Instead you should be installing on top of some sort of concrete floor. You could either add mud/self leveling mix or cement board. You may also want to remove the hardwood first, if you can.
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Generally you would put the trim on after the tile is done. Especially if you are laying ceramic tile and need a mud base or cement board. If the room connects to another without tile, I would do the trim first. For example, the kitchen has tile and it flows directly to the dining room that has wood down and won't be replaced. If you put the trim down after the tile, the trim in the kitchen will be at least 3/4" taller then the dining room. This is only an issue when you have a really open floor plan.