Edit: I see you have called a professional. My apologies for misreading your question.
It is not unheard of to have different contractors tell you different size wires- some will go for the minimum, some for future proofing. The cost difference between 10 gauge and 12 gauge is minimal, especially for short runs (other than being more difficult to bend). Different contractors may also know what the inspectors like to see- so if they're expecting to see a 10 gauge wire (30A), and the contractor installs 12 gauge (20A), they'll get burned.
With that said, absolute minimum is a 12gaug or 12/2 (so Hot-Hot Ground) However that assumes everything is 220V, and if the system has 110V anywhere then you might need a neutral. I don't know this particular unit, but a photo of the wiring diagram in the manual will help.
The run from my AC is just under 10'. It should be protected in conduit outside the house, run to a disconnect box (fused at 30A), and from there via flex into the compressor.
The cost of the wiring is negligible. The cost difference of the breakers should be 10$, maybe 20$ with markup. The labor is the expensive part.
Assuming they are running a new lineset these costs should be the same. The manufacturer says the size of the copper lines, and copper is copper. If they say 20' of 3/8 you can look up the base price of that- and everyone should be scaled by the same amount (Do NOT think you're going to get copper lineset at store prices, not going to happen).
Overall a bit of future proofing should be useful, but without more details- distance, quotes, environment (do you need a new pad, etc), and what after-the-install service is, prices can vary.
There are websites you can go to to get an estimate as to what things cost in your community. They're distributions, obviously, but it may help.
You probably do not want the cheapest bid, especially if it's by a margin. Calculate the median, average, and std. deviation of the prices. Always a good metric...
You should call a professional.
4 individual contactors? Typically there's a compressor and fan, and then an 24v AC that turns them on.
You need the manual which will tell you the minimum sizes, wires, and whatnot, and you'll also need to know local code for what they need to be run to and any external or internal disconnects you have to provide for service.
Do this wrong and you'll be lucky to end up in the hospital.