A few things -
You said you got the dog last March so the dog should be old enough to hold it over night. Also, he is going in the same spot on the rug because it smells like pee there and he likes it.
To clean the rug you need to probably steam clean it and use an odor neutralizer specifically for dog pee (Nature's Miracle, etc).
Most of all - sounds like your dog needs to be crated. You said he's crate-trained, but are you using the crate? He shouldn't need to go out in the middle of the night, so if he needs to, you should maybe see a vet. It could be a UTI or something.
Also, I would read the Sticky Articles here in the dogs section to brush up on potty training. You should be taking him out on a leash, etc and praising him when he goes outside (like a puppy).
Cool job on getting a shelter dog!
Hope it works out.
Let me/us know if you need more help.
I'm sure that others have more advice and remember something I've forgotten.