Bathroom remodel. Demo down to bare studs and 1" x 6" subfloor. Do I put plywood or anything down on subfloor before I install cement board? and do I put anything under tub?

IF your joist spacing is on 12 inch center or is of something greater than 2x6 joists, you could 'get away' with laying cement board directly on the floor.

Minimum code usually allows for that. Minimum is the key word and should tell you a lot.

I personally prefer to beef up the floor with an exterior or marine (water resistant) grade tongue and groove ply, at least 1/2 inch - I would most likely go with 3/4 inch because I like to over build.

You figure 3/4 inch ply, 1/2 inch cement board and 1/4 inch tiles you are talking a build up of roughly 1 1/2 inches - it isn't that much, however it gives a bit of extra bulk and provides a layer of protection against accidental flooding. It is also somewhat more stable it doesn't give as much as a single 1" material. When laying tile you want the floor to not move.

There are new products out there designed to make floors water tight. Most are plastic based products, one is specifically designed to be laid underneath tile flooring, it comes in a single sheet, is laid down with adhesive. Sorry I forget the name of the product. I think its called a membrane...

http://www.hgtvpro.com/hpro/nws_econ_fin_material/article/0,,HPRO_26526_5684288,00.html may have some helpful information.

Under the tub:

If this is a fiberglass insert, with a front, you will want to insert it in such a way that the bottom lip is level with the cement board, or resting on the cement board. You will also want the bottom of the tub laying on the 'floor' of your tub space. Usually a single sheet of ply laid in provides the added bulk, and provides a single flat surface.

If this is just a tub insert, like a jacuzzi, then you need to follow the manufacturer's instructions on support of the floor of the tub, of course consideration for access to the additional pumps and plumbing (more commonly on through the floor on first (ground) floor application where there is a cellar or crawl space, through an adjoining wall on second story application.

Anything else (steel, iron) you most likely will want to beef up the joists and more or less lay in enough extra material to level and raise the tub to meet the 'new floor' level.

Of course laying in extra layers means you will have to adjust your toilet collar and your tub drain.

There is little strength in CBU, cement boards. If you are going to tile the floor, you need to check the deflection of the floor, and it will depend on your floor thickness, joist size and spacing and other things. You can check at this website, just plunk in the numbers and they will tell you what you need.
http://www.johnbridge.com/vbulletin/deflecto.pl

It's a very good website for questions about flooring so if you need to know more, try them. It's full of the best help online anywhere, really wise, friendly people. They know a lot more than just tile.