He has one heart.
It's what canon gave and what I choose to stick with. One heart and that's all. He will never be a Time Lord, never even be close. He doesn't even have the chance of having that sort of mind.
He isn't human though, he can't be. I can't see what happened to Donna Noble, how she changed through the process and gained such a powerful mind but the Doctor lose his biological identity. Yes, it's not the Doctor completely but that hand was a part of him. That hand was a part of a full Time Lord and, in my mind, I cannot understand, even with Donna's influence, how it can grow into something completely human.
So that's where he comes in.
He's not human, no human has five regenerations and no human has the residual time sense that he has but he's no Time Lord either. He does not have the mental capabilities they do and while that sense of Time does exist within him, its faint, almost nonexistent in comparison to a normal Time Lord.
He does not look at something and see what needs to be or what it once was, he simply sees what it is.
However, also like Donna, he does not exist easily. All his senses are dulled, with touch being the strongest and his body is weak. Even his mind is impaired, severe memory issues constantly plaguing him and even on his best days, there are things he easily mixes up or forgets all together.
He will never, for example, be able to fly a TARDIS.
However, his body does have the ability to mend itself to a minor extent but it requires him to go into an almost comatose state to do so and this is usually required about once every six weeks to keep him functioning on a decent level. He can feel these states coming on, his memory becomes even worse and his motor skills steadily decrease until he's gone.
These periods, depending on how hard he fights against them and what he has been doing in general, can last anywhere between two days to two weeks.
As stated earlier he has five regenerations. Everything that has been discussed above applies to his first. Each regeneration will see him steadily becoming worse until his final regeneration which will leave him not only fairly useless but in an extreme amount of pain and possibly a bit on the mental side of the spectrum. The regenerations are not smooth, not even close and, for him, they are severely painful and rather traumatic.
In short, he tries to avoid doing it at all costs.