Car began to backfire after engine gaskets changed. And radiator water overhears and steams out of cap.

Just had all engine gaskets changed plus block was rebalanced (if that is correct term). This is 1st time done in cars 12 yrs. It all started when water in radiator began to overheat and steam out through radiator cap. Was told by mechanic this was due to leakage of pressure due to worn gaskets in engine. So on his advice I had the job done. While dismantling the engine he found there was a leak from radiator at bottom right connectors. Had it pressure checked and no leak found, but when he again replaced it, after engine was replaced, the radiator leak grew. So I bought a new one, which was 1st time also, had it intstalled, but THE SAME CONDITION IS STILL OCCURING. I spent alot to have this engine done, because supposedly it would correct the radiator water from over heating problem. But it hasn't, not even with the new radiator. So if the worn gaskets and supposed lost of air pressure from engine was causing this, what is still happening, now that the gaskets are new? And engine rebuilt?
Additionally the car now backfires and the mechanic keeps passing the buck! First he said it was due to radiator, had it replaced, now he says it due to computer codes needing clearing, checked that, scanner man said NO! Then transmission was leaking, it too needed new gaskets, 1st time for them too. By the way this had been a great car, very low maintenance or servicing, very lucky. So now what? I told him to clean and recaliper the sparks plugs, and generally clean and check everything, which I hope he did!
I must tell you that I live in Mexico, and there are no modern anythings here! They do most things the old way, especially in more modern computerized cars. They don't scan prior to repairs, its costly to have done. After repairs there is no clearing up and reprogramming of codes either! Most scan owners don't know what the codes mean or refer too. So it's total Russian roulet.
I have read on your site causes for backfire and they all relate to the engine in one way or another. What has he NOT DONE or should have done that is causing the backfires. What must I tell him to do so he can finally repair my car to its former smooth, quiet running self? If it could be translated to SPANISH, all the better.
So my two major concerns are 1. Water overheating in tube just below the radiator cap and steaming out. Cause and repair 2. New backfires once I give it gas, unless I let it warm up for 5 minutes. It feels like a momentary vacuum at which time I must reaccelerate, and then I hear the bang! Sometimes it occur after slowing car down and giving it gas to proceed, vacuum, then 2 lighter rapid explosions. This all started after engine job! My car had NEVER backfired before this!
I await you prompt response and sorry I can't donate have a
Ready spent a fortune and car is still nor running. Patria