Lots of different subjects discussed here and not sure where you stand on the electrical check out. But a few thoughts on simple check out of electrical.
What you can do electrically before going to diagnosing the rest of the A/C system with gauges or a professional.
1. With tractor switch on but not running and A/C switched on, put a voltmeter on the lead going to the clutch on the A/C compressor. If you have 12v, the clutch is getting power.
2. If clutch is getting power, turn off A/C, start tractor, turn A/C back on, if voltmeter shows 12v, does the clutch engage and turn compressor? Is there any sign of compressor trying to turn? Does part of the clutch jump when power is turned on? If not trying to turn (look at compressor shaft to see), the clutch is probably bad. Shut off engine and A/C. Disconnect wire to clutch terminal, and set voltmeter to OHMs LOW range. Connect one voltmeter lead to terminal on clutch and other lead to the clutch body. (Clean an area to get a good contact). The resistance should be less than 100 ohms. If it is infinity or 0, the coil windings in the clutch have failed and it needs to be replaced. If the clutch is trying to turn, or part of the clutch moves when power is on, the compressor is bound up or there is high refrigerant pressure in the system. You will need an A/C mechanic to test the system to see if there is over pressure, check the compressor and clutch and repair as necessary.
3. If the clutch is not getting power you can do some more electrical testing. You will probably need a wiring diagram.
4. Does the power from the relay run directly to the clutch in the diagram? If so , is the relay turning ON? If the relay does turn on (12v at the outlet power terminal on the relay, usually terminal 30) follow the wire to the clutch. Any damage or loose connections to repair?
5. If relay is not turning on, is the fuse good and does it have input power?
6 . If relay has input power, check the control input terminals(usually terminals 87a and 87b) One will be hot and one will be ground. The control switch for the A/C is usually on the ground side. Look at the wiring diagram to see if you can tell whether the switch is on the hot or the ground side. But you can test it without knowing which. Turn on ignition but not A/C, set voltmeter to 20v range. Connect meter red wire to a small probe(small screwdriver), connect meter black wire to a ground on tractor or battery. Use the probe to see if either terminal 87a or 87b in the relay socket is hot. If one is, the switch is on the ground side, and power to the relay control circuit is good. If neither terminal is hot, turn on A/C switch and see if either terminal is hot. If it is, the switch is on the hot side.
7. If switch is on the hot wire, or if you get no hot connection at either terminal, put probe in negative lead of meter. Put meter in 20 v setting and put red meter lead on battery positive terminal. Put probe into other 87 terminal of relay socket. You should read 12v in one. (The switch is on the hot side) If not, check the wire to that terminal and make sure it goes to a good ground.
8. If you get a good ground on one 87 terminal and no hot at either terminal with the A/C switch on, the A/C switch is on the hot side of the solenoid control circuit and the ground side of the control loop on the 87 terminals is good. If you don't get a hot or a good ground at either 87 terminal, Check the wiring diagram to find which side the A/C switch is on and follow the wires to see why the other 87 terminal is not getting ground or power.
8. Finally, to check the A/C switch and high/low pressure switch, follow the wire from the A/C switch to the 87 terminal it connects to. Somewhere along the wiring it will connect to a High/Low pressure switch(Usually small round with 2 terminals) screwed into a fitting on an A/C hose or evaporator. Set A/C switch on, Ignition on. First use the voltmeter to check the A/C switch. Set for 20v, connect positive lead to output terminal of A/C switch(terminal which wire to solenoid 87 terminal from switch is connected to) touch negative lead of voltmeter to battery negative. If 12v, switch controls positive side and is good. If no reading, connect voltmeter negative to battery positive. If 12v, switch controls negative side and is good. If no reading on either battery terminal, switch is bad, or its connection to power or ground is bad. Check fuse and wiring to other side of the switch. Make sure it goes to fuse or ground.
9. Check pressure switch. You have determined if the A/C switch is on positive or ground side. Follow the wire from the A/C switch to the pressure switch. Leave that connected. Disconnect the wire from the the pressure switch to the A/C clutch. Using the same voltmeter connections(First battery positive and then negative) and probe, turn A/C switch on. Touch probe to each of the pressure switch terminals . If you get 12v. reading on both terminals with the A/C switch on. the pressure switch is good and not reading too high or low pressure. Check wiring to solenoid 87 terminal from switch. If good, compressor clutch is probably bad. If you get 12v at only 1 terminal on the switch, the switch is shut off or bad and you will have to get an A/C mechanic to find out why. If you get 12v on neither terminal, check wiring to pressure switch from the A/C switch.
If you want to do a very quick test, without working through all the steps, find the high/low pressure switch. Disconnect both wires from its terminals. Set voltmeter to OHMs LOW range, and connect the leads to both terminals on switch. If ohms are 0, switch is good and system is not over/under pressure. If ohms are high, switch is reading over/under pressure in system or it is bad. You will need an A/C mechanic to fix system or replace switch. If switch is good reconnect wires to terminals, turn ignition and A/C switch on and check clutch terminal with voltmeter set to 20v.. If 12v reading, clutch is bad. If 0v start wiring check above and you will find something wrong.
Hope this helps.