Does the thickness of insulation on the cooling pipe of an air contioner make a difference to preventing mould and condensation

Check your start / run capacitors. They live in the outside unit, near the switch.

They look like oval or circular cans, and they feature prongs on one end and a flat bottom on the other. If either end looks even remotely non-flat, the capacitor has failed and the compressor won't run (even if there's freon in the line and the compressor is functional). Sometimes there is only one capacitor, as it has both the "start" capacitor and the "run" capacitor built into the same package.

If you are in doubt as to the "flatness" of the package, pull the disconnect near the outside unit, provide a bit of time to let the capacitors drain (10 min should be sufficent) and pull the capacitors out of the system (take care to identify the wiring, wiring a capacitor backwards is not recommended). Then find a nice flat table and if one wobbles, it is gone. If it is flat on the table, you dont't have this problem.

The repair (if needed) is a new capacitor ($10 to $15 at an AC parts supplier, no license needed for this part). If you find these capacitors failing more often than "once in a blue moon" it is a strong indication that something else is wrong with the system, like worn windings on the compressor motor, or possibly an electrical short elsewhere.

You can replace a two-in-one capacitor with two single capacitors (provided they match the original ratings) and vice-versa. But there's no need to get all fancy.

And everyone says "low freon" or a compressor. Get used to it. Occasionally they are right, but a failure like that sounds odd if recent repair work was done, unless the repairman did a shoddy job.

It sounds like you are getting very questionable advice.

Wet drywall needs to be cut out and replaced. Damp drywall is a prime source of mold. Cabinets that are bubbled halfway up have major water damage and may not be easily repaired.

You should consider getting an independent inspector to assess the damage and the repairs needed. If their view is significantly different from the position of the plumber (or his insurance company) you can try to get them to pay for the cost of inspection.