Does the doctor have any kids?

yes he did, In the episode The doctor's daughter-series 4- the doctor mentioned to Matha and Donna that he once had children. nobody knows their names yet.

He does now have a New daughter Jenny who he got from some skin on his hand, A solider forced the 10th doctor's hand into a progenation machine. A tissue sample is extrapolated and accelerated, and within minutes the Doctor becomes a father! again.

For more detail my visual dictionary gave me more complex Words but the rest i knew myself.

He also has a grand-daughter Susan.