We plan on very tiny joints. Are spacers used for very thin grout lines?

Yes. Usually they're small tapered spacers called "marble wedges", as seen in the pic below.

but Bill Vincent recommended siliconized latex caulk in another thread. Are there particular brands that perform well?

Whatever the brand grout you used, that's the brand of caulking you get, as well, and from the same place, and in the same color as the grout. Just about all grout manufacturers now make their caulkings with antimicrobial agents in them to put a stop to mold and mildew.

Do we fill the tub with water before caulking?

If it's cast enamel, it's not necessary. If it's plastic or fiberglass, you'd do alot better to make sure the installer supports the "belly" of the tub with mortar or foam. It'll go alot further than filling the tub. The reason for filling the tub is so that excessive movement won't stretch the caulking past its limits. Even a better idea is to limit said movement (which is what you do by supporting the bottom of the tub).

Also, I remember a thread mentioning using caulk instead of grout in the "corners," but I am not sure if that means just the bottom corners, or the whole vertical seams where the back and side walls of tile meet.

Any place where one wall meets another, or a wall meets a floor (or in this case the tub), it should be caulked.

Thanks, Bill. I went back to the tile store where I bought the tile and grout and asked for siliconized latex caulk, but they do not carry it. The grout we bought is Durabond in white (there was only one shade of white).

I have to go to Home Depot to try to find siliconized latex caulk in white, and the odds of finding Durabond are low. Are there other brands you would recommend?

At Home Depot, go to their tile section and get the one from Custom Building Products.

As for the Durabond, their sister company, Hydroment, also has colors that would be an exact match. Buf for just plain white, any grout company's caulking will do fine.