Time Lords have not usually had children in a normal fashion for a while.
According to one account, all pregnancies on Gallifrey ended in
miscarriage when the Pythia cursed her world with sterility. Even
Rassilon's wife was said to have endured this curse. (PROSE: Cat's
Cradle: Time's Crucible) This required Time Lords to adopt the asexual
"Looming" process of artificial genetic reproduction. This resulted in
the elimination of direct, lineal familial relations, rendering all
relationships collateral. In other words, Time Lords of the same
family (or "Chapterhouse") could at most be only cousins of each
other. However, the Pythia's curse did not seem to extend to
off-worlders, as Leela was pregnant with Andred's baby. (PROSE:
It's possible for The Doctor and River Song to have a child, and it will most definitely be part Timelord. Now if the child will inherit certain traits of Timelords is more difficult to determine, I do not believe there are any legit half Time Lords in existence other than the Meta Crisis Doctor.
The Meta-Crisis Tenth Doctor was a Time Lord/human hybrid and a human
incarnation of the Doctor, who possessed one heart, aged as humans did
and had no regenerative ability to avoid death. He was created by an
instantaneous biological meta crisis and was the final result of the
Tenth Doctor's partly sidestepped eleventh regeneration.
This shows that at least in this scenario of a hybrid he doesn't have any distinguishable traits of a Time Lord.
As somebody stated there is an interview where Moffat talks about The Doctor and River having a kid.
Q: “Why does River and the Doctor have babies?” (laughter)
Matt Smith: “That’s a very good question.”
Steven Moffat: “Why don’t they? Well, who’s to say what goes on? We
don’t see every chapter of the Doctor’s life. But heaven knows what
they’d be like. Can you imagine those two as parents?”