Can You Install Tile Over Plywood?

Tile adhesives do not stick very well to wood and will usually let loose and pop off over a short period of time.
Suggest you pick up a piece of backer board from a home supply store, ( cost around $7.00 for a sheet), glue or screw it down to the wood, then tile over the backer board.
  Backer board is only about 1/8 inch thick and easy to cut. Just score it with a razor knife and pop it apart.
Suggest you sink the screws into the backer just below the surface so that you have a smooth surface to lay the tile. You can use premixed tile mastic to adhere the tile to the board with.

Yes you can, but you need to ensure that the surface is level or the tile will crack. Also ensure that the plywood is fastened down with screws and not nails as overtime the nails will come loose and break the tiles. The surface of the plywood should be clean.

Have two fit mosaic tiles on mats as a border between 12mm thick tiles mosaics are 6mm thinner than ceramic tiles how do you bed them out 6mm and not have them drop

Yes.  Although it is not the optimum situation.  Make sure there is no bounce in the floor (deflection).  Floor must be clean, flat, and suitable to receive thin-set.  I would use a latex modified thin-set and primer the floor first with a primer recommended by the thin-set manufacturer.  This will help the thin-set stick to the plywood. 

I need to tile over existing plywood bathroom floor. What type of waterproof membrane should I use vs backer board?

Both of the other two are wrong check the plywood secure it for no movement then you put down backer board don't use nails  then thinset tile grout seal grout then doon.thats how I have all may employees do it for the past 15 yrs.

Why not try painting the floor instead with durable floor paint. Dulux have a range but have a look around for other manufacturers too. The effect is modern and fresh, the paint had a rubberised, durable feel perfect for the bathroom. Brother-in-law has done this and it looks great. We're going to paint over the tiles in our kitchen with it to update the look cheaply!!

Absoulutly not. The grout line between the tile will remain pouris and water will be able to seep through. You will either need to lay down a water proof membrane down first of backer board such as duro rock.

First you must check to see if the plywood needs to be re nailed in places to prevent movement or squeaking down the road. Once you are sure of the integrity of the sub floor, there are adhesives to purchase to use to secure the tile. Grouting is last and there are choices of color if you want.