You would usually caulk the gap between the baseboard and the wall. This is why I usually don't bother painting the trim before putting it up (unless I'm staining it).
Usually I would prime the wall first, then put up the trim, putty/caulk, paint the trim, then finally paint the walls.
The easiest way to remove things is usually in the inverse order they were put in.
If you intend to put down new flooring, you must remove the tile anyway, so remove the tile first. I'll admit that the scariest thing to do might be the tear out. It forces you to accept that you are doing this thing, especially if you have never done something of this magnitude. It will make a mess of things for a wee bit.
I would suggest removing tile with a hammer and wide cold chisel. Break up at least a few tiles by hand. If you have a compressor available, a great tool here would be a flat air powered chisel. This tool would rapidly allow you to peel up the old tile. A bit noisy, but fast. If not, then the hammer and hand held chisel (don't use a wood chisel unless you have borrowed the wood chisel from someone you don't like) will do. Once a few tiles are up, you may find that a large ice scraper, used for breaking the ice from your sidewalk, will be useful to peel up the rest of the tile.
Once the tile has been removed, then remove the baseboard for later replacement.
Demolition can be the most fun part of a project. Try to keep the dust out of the remainder of the house, and use a dust mask.
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I don't think that's a very good idea. First of all, the wood is already sealed, so the paint isn't going to adhere very well. Second, you're going to waste a lot of money by purchasing nice grainy oak baseboard, since you're paying for the color and grain that are going to be covered up by paint. You can save a dollar per linear foot or more by going with some finger-jointed pine or MDF pre-primed white baseboard that you then finish off with a coat of paint.