Plywood is the best choice between these two. Particle board can swell if water gets to it. You would be better off if you use a tile backer board. Concrete based or a fiber type. They are made for this application.
No, plywood alone is not strong enough for ceramic tile. These floors require a very sturdy subfloor at least 1 1/4" thick before the tile. This requires either using cement board or a pored mudbed.
The best way to repair particle board sub-flooring is to cut out the bad spot and replace with new board. You can only use vinyl and self stick tile over particle board. Ceramic tile cannot because the moisture in the thin-sets will break down the particle board.
You couldn't tile directly over a plywood sub-floor because over time, the wood will be expanding. There are products you can use in between the plywood and tiles to keep them in place such as cement backer board.
No because of a few of reasons. Most manufactured homes have particle board subfloors which are completely incompatible with installing ceramic flooring. In the rare event your floor doesn't have particle board and does have plywood, the house must be on a foundation. If it is in a park, it is usually held in place with tie downs. The home moves to much for the floor to be successful. Finally linoleum isn't an acceptable subfloor. You need a total of 1 1/4" of subflooring when complete. Assuming you have 3/4" plywood, you would need to add 1/2" cement board. The linoleum can stay, but the tile cannot be adhered directly to it.
I don't recommend it. The plywood is flexible and bends over time. Cover the plywood with Hardi Backer board first. Available in the tile department at Lowe's and Home Depot. Get the right screws for it. It's easy to install, comes in manageable size of 3 feet by 5 feet. Well worth the time, effort and money to make sure your tile job lasts.
Improper installation. You must install cement board over the sub floor before you tile. If the tile was glued to plywood which is not waterproof, the plywood will absorb moisture which will cause the grout to crack and deteriorate
Hardwood floors are generally more expensive than tile floors because they are harder to maintain. However, higher quality tile floors can cost more than hardwood floors.
That depends upon location. Tiles can be installed on a gypsum board wall or ceiling. (But make sure that the tiles are well grouted; if moisure gets past the tile into the gypsum board, it will be damaged.) Floors require greater strength - but then, you wouldn't normally have a gypsum board floor, with or without tile.
This is not a good idea without using a cement /mortar board underlayment.
It depends on what you are gluing it to (concrete, plywood, tile). Research the different adhesives available from your flooring dealer.
You sweep the floors
Wonder board is a cement board that is very firm. Unlike wood it doesn't warp or rot. It is especially used as the underlayment for tile floors and shower walls.