Not only is using # 10 wire on a 60 amp breaker a terrible idea, but it is also illegal. The 60 amp breaker is also absolutely superfluous. A 30 amp breaker (appropriate for # 10 wire) is sufficient for this compressor.
The best way to find out if your wiring can handle more than 20 amps is to turn off the power and check for hot wires. If they're all green or black, then you know that your wiring is capable of handling more than 20 amps. If one or more are not, then you should probably stick with 20 amps until you can replace the wiring.
In the US, most houses are built with 20-amp circuits in mind. Most appliances are designed to work with 20 amps, so there's no reason to use anything else. If you need a bigger circuit to run a air conditioner or heater, you can always install a 30-amp breaker.
It's important to remember that greater amounts of current mean higher voltage levels too. So even if your house is wired for 20 amps, if you have any receptacles on 20-amp circuits that are still plugged in from last time you had electricity, those outlets will see a high voltage when the power comes back on after a storm. You should never plug anything into a live receptacle - not even a dummy load like a radio or lamp - unless you want that thing ruined forever.
No, 10 gauge wire is not designed to carry 40 amps. You want to keep your 10 gauge wire at 30 amps, but for 40 amps, you'll want to use 8 gauge wire. Furthermore, 12 gauge wire can handle 20 amps and 6 gauge wire can handle 55 amps. Is it safe to use a 20-amp breaker with 10-gauge wire? Yes, as long as the total weight of all the circuits does not exceed 100 pounds. The metal parts of the house are usually made from steel or aluminum, which are both heavy metals that are not affected by heat or cold. But if you install circuit boards or components into wood or fiberglass furniture, they may get hot and burn those things down. It's best to use protection devices to prevent excessive current from flowing through these small wires.
10 gauge wire comes in lengths of about 250 feet, so it's easy to buy enough for your project. Each junction box should have one conductor per circuit, except for the neutral conductor, which can be shared by multiple boxes. The other end of the cable needs to be connected to the proper terminal of a feeder bushing or conduit connector. The final step is to connect each conductor to the appropriate voltage pole. This can be done using wire connectors or tape.
If you're just wiring a single room, then a 15-amp circuit will be sufficient. If you're adding lights, appliances, and other equipment to a single room, then you should run a 20-amp circuit.
"Twelve-gauge wire can handle 20 amps, 10-gauge wire can handle 30 amps, 8-gauge wire can handle 40 amps, and 6-gauge wire can handle 55 amps," and "The circuit breaker or fuse is always sized to protect the conductor [wire]."
So if you're wiring a house, unless you have specific requirements for your circuit conductors that only 12-gauge wire will not meet (which is very unusual), use the largest size you can find that still fits in your conduit.
Here's something to consider: If you were to connect multiple houses together with their own circuits, would they all need 12-gauge wire? Probably not. 11-gauge would probably be sufficient to connect two houses together. The reason being is that when you connect two pieces of wire together, even if they are from different houses, they will usually share the same voltage until you get down to about 3 or 4 volts where it becomes too low for the other wire to function properly. So as long as the total voltage across all the wires in each connection is less than 12 volts, they should be fine.
Now, what if one of those connections is going to be a light switch? Well, in this case you'll want to use 14-gauge wire because lights typically require a minimum voltage of 120 volts to work.