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.
In the to-do section of the care instructions you posted, it suggests "Use an all-purpose, non-oil based household or commercial cleaner." OR, it says, you can use vinegar/water.
I think you are over-analyzing. The normal floor mopping solutions like Pine-Sol or a Swiffer will work fine. Just don't use anything abrasive that will scratch the tile or an aggressive bathroom "tile/grout cleaner" that is meant to bleach and renew grout that is stained.
If you're concerned about a particular cleaner, you could always test a small section that is out of sight. You can also apply a "grout sealer" solution over the grout lines. This is tedious, but can help slow down staining of the grout.
Best Answer
It would be easier for people to see what's going on if you move the rug to photograph the whole affected area. It's impossible to tell the extent or location of the crack.
Do you have a spare tile, or can you buy one? Do you know the cause of the crack (like something being dropped on it)?
With better information, we may be able to give you a better answer, but with no other information, I would guess that either the tile wasn't properly supported, or it was installed on a surface that flexes. Without fixing the cause, there is no real repair that will last.
Once the support surface is solid, the only repair that will look good will be replacing the tile, which is also the easiest and fastest solution.
If you can't find a matching tile, an artist may be able to mask the crack by painting filler to match the tile's pattern. That could be an improvement, but it will still be visible and won't hold up to wear.