“God went to hell?”
Yes, he certainly did.
“Then would not death be an example of mutability? Of change?”
So is the incarnation.
“That flesh and blood human nature is in heaven?”
Indeed. For the resurrection is of the body, cleansed and purified and incorrrupt.
“But again how can God gain (or now have a human nature in addition to the divine nature) and retain it if He is immutable?”
By choosing to become man. The incarnation isn’t a temporary cloak to be tossed off after death. It was permanent.