Almost for certain, you have water damage in your sub floor which is weakening the floor, causing it to sag, causing the tile to crack. Yes you could replace the sub floor and flooring, but find what's causing the water damages and fix that first. or a joist could be broken
under a foundation of a sub.
Usually yes, a well insulated sub-floor will save you lots of money from heating. You can choose to build a sub-floor on 2x4 lumber with moisture barrier or use those sub-floor tiles (usually come in as 2x2).
about 3.50
depends on how much water, but in all probability, yes
to support the sub floor.
Dricore
The type of underlay you need will depend on the type of sub floor or floor you are putting the laminate on is made of. If the sub floor is made of concrete, you would want vapor barrier underlay. If the sub floor is made of wood a standard or premium underlay would used.
Absolutely.
If you're installing hardwood floor boards over a strong 3/4 inch plywood sub-floor, then yes, you can install the hardwood floor parallel to the joists. If your sub-floor is weak, then you should fix the subfloor.
Yes, a laminate floor will do just fine in your kitchen. Unlike wood floors, the laminate is very resistant to damage from moisture. Be sure the sub-floor is prepared properly, however! Sever water damage to a sub-floor can cause warping and buckling if the proper materials are not used. If this happens, damage to the sub-floor will telegraph problems into your new laminate floor, causing it to warp and buckle as well. Hire a qualified contractor to install your floor who knows what he's doing and who follows all appropriate building codes. But a sloppy dog probably isn't much to worry about. :)
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.