How to remove stubborn nails from baseboard

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.

There needs to be a 1/4" gap next to all walls to allow the flooring room to expand.

This gap should be covered with trim of some sort. The problem with baseboard alone, is that most baseboard is 3/8" thick, leaving only an 1/8" of an inch for the wood to contract without exposing the edge. This also assumes that the flooring is laid with tremendous precision, which it is likely not.

So your options are:

1. Find thicker baseboards. Door and window casing comes in 3/4" thickness and can be used.
2. Use quarter round or shoe molding to thicken the baseboard.
3. Use door stop in the same way you would use shoe molding.
4. Undercut the drywall so the flooring can expand underneath and use regular baseboard (thanks Matthew)

I had the 3rd option done in my basement, and it turned out pretty nicely.

I would also recommend the caulk. It makes everything look nicer.