It certianly sounds like the old HDD is faulty.
If it were me i would set the re-built pc to use the new HDD, with whatever OS on it that you choose.
Once it is all up and running I would put the old HDD in as a either primary slave or secondary slave.
and TRY to copy the files you want off.
But if the old HDD is faulty there is the chance that the faulty drive might to damage to the new mobo. It depends on what the fault is, and that is anyone's guess.
Other than that it is a data recovery service/program, and i will leave it to others to make suggestions, as i have no experiance in this area.