The computer doesn't control the heat, the coolant flows to the heater core at all times (so long as the thermostat is open, I guess?), but the fan & vents don't route through the computer.
OP, I support you in your efforts to de-computerize. The long-term availability of the computer is something that worries me, but I'm probably old enough that I'm not going to have to really worry about it. Anything is possible, so I'm sure converting your 4.0L to carb fed can be done. But to make it something you can do on your own you're going to need to understand everything that the computer controls. First and foremost in my mind is ignition. Your timing is set by the computer, so you'll need to find a distributor that will fit a 4.0 but has a mechanical or vacuum advance and allows for the initial timing to be adjusted by hand. Next is fuel delivery, because honestly I don't really know how the fuel delivery works in our TJs, but all I can think about is how every carb'd car I've had also had a mechanical fuel pump mounted on the block, which you won't have.
The gauges won't be that hard, you can run aftermarket oil pressure, water temp, and voltmeter for sure. Fuel level might be a trick, I'm not sure what magic the computer performs on it before it gets to the gauge, but maybe nothing? Or maybe there is a known ohm range for our fuel sender and a specific fuel gauge might work, idk. Speedo can be run directly off the existing sender & the tach can be fed from the #1 spark plug wire. The A/C could be tricky, since the high & low pressure sensors tell the compressor when to kick on/off. You're going to be doing a bunch of custom wiring no matter what.
Your 42RLE transmission probably has to go, it's computer controlled. But you said you wanted a stick anyway so problem solved.
Dropping in a 350 & matching trans is easier insofar as it's already prep'd for being carb fed & lo-fi ignition/timing, but you introduce new problems like motor mount locations, transmission mounts, rear driveshaft length & pinion angles and probably a mess of other things. Oh, and the 350 typically requires a different radiator because of how the hoses lay out. You need one with the inlet/outlet on opposite sides from where they are on the TJ. But that road is well-traveled so you will be able to find a bunch of resources to help you there. And probably a thousand other things.
The availability of the computer it one thing, have you tried to get a decent carburetor or rebuild kit lately? New carb would be the way to go. Fuel pump remove assembly from tank fit a low pressure pump to existing assembly, buy fuel gauge to match factory sender values. A/c is no problem. I think you could just attach the a/c input wire to the A/C on wire together at the computer. May need a relay.