Yes, this is normal behavior.
You wrote 'When fans turn on, they have to bring the temp down and turn off and this cycle should go on and on...'
That is not completely accurate. There is a set upper temperature limit and a set lower limit for each fan speed mode. When the temp hits the upper temp limit for the low speed fan operation, the fans will turn on ...
Drilling holes big enough to allow a socket through would weaken it a lot. And as you don't want to weld new nuts on then you would not be welding up the holes either....
I suggest you do it properly or get it done properly and rivnuts are not strong enough.
Here are what these codes mean:
P1336: P1336 Description
P1339: P1339 Description
I cannot imagine any connection to the fuel injector. It's best, IMHO, to assume that the codes are accurate and follow that until you prove otherwise.
Most likely you have some wiring problems. Find a better mechanic!
More than likely when the engine sat for so long, one or two of the cylinders (open to the elements via an open intake or exhaust valve) became rusted. When you turned it over, it forced the piston rings up into the rusted area of the cylinder and now it is stuck.
You have two options, depending on your pocket book.
Take what's left of the engine out and ...