Did all this electrical hooey start immediately after the transmission was installed? Or a few weeks later? If it didn’t happen immediately, it is pretty much impossible for the transmission to cause any of this. If it happened immediately then it is possible the installer didn’t reconnect the ground wires.
So we know it is a 5.3 V8 Tahoe 4WD but know nothing else about it. Mileage? Did you recently buy it? Was it a flood car (salvage title)?
Since you don’t sound like the DIY type, take it into a good independent mechanic (not a chain) and expect to pay for fir his time finding all these problems, let alone fixing them.