Do you know what kind of adhesive was used? It seems to me that the first step is to break the tile from the backing and then to cut out as much of the backing as possible to give close access to the joint between the cabinet components and the backing. Then apply heat (maybe an old iron) to the plywood backing to soften the adhesive and pry the backing from the cabinet components. This is a tedious process, but it should prevent extensive damage to the cabinets.

If it's indeed glued, u probly have no choice but to demo the wall behind and install new. Drywall is not that hard to do. I have a vow after I bought my first house, any contractor who exhibit annoyance, not helpful in answering questions, look down at job as too menial, will take care ONLY of certain things but not others, are to be shown the door and please don't call.

The glue they use to adhere the top to the cabinet is stronger joint than what is there in the top construction. Simply buy a demo crow bar and hammer it in between the blocks and counter NOT the block and the cabinet. It will easily break free. Particle board is weak and will splinter into pieces. If you have a saw you can cut out the center of the counter giving better access to the blocks. Also you would be separating smaller sections. Fix your valve you can flood your entire home.

The reason the person who installed the pipe to wrap around the condensate pump was to create a P-trap. It looks ugly but it works. The problem is everything is glued together and if a clog develops you won't know about it until it is too late and you have to cut the pipe. I second the recommendation of adding a clear EZ Trap. You will see the clog develop and it is easy to open and clean. On some models a clean out brush is supplied. Usually coil cases have two threaded connections. Yours has a third unthreaded connection so a PVC can be glued into place. The drain should be at the lowest point. The second connection allows you to install a condensate alarm. This is wired so that if the drain is clogged the water level rises to the second connection and the alarm shuts down the AC. This prevents a flood and forces you to clean the drain. EZ Trap also makes this condensate alarm.