wheel swap

I’ve read the threads on using other rims/tires from other vehicles. I am not familiar or understand all the measurements to figure out what will work without modifications. I don’t want to get something and it be wrong or need work I can’t do. This is my sons truck and the current rims/tires...

I ended up trading in my '03 for a similar '10, but I didn't care for the P235/75R15s on the '10. So, after being told by dealer 'no problem', I had them swap the LT245/75R16s onto the '10. Mechanic is telling me he calibrated it as much as he could, but my speedometer is off by 4-5 MPH. Instead...