Musings on Who: Does The Doctor Age?

For the past few days, my whovian friends and I have been grappling with the question: does The Doctor age? Without (much) research, here's my thinking on the subject:

1) The first doctor is old!

He couldn't have just popped out an old man! His baby carriage is shown in one episode, and in another episode it actually shows him as a child (And his face is far from that of a grandpa)!

2) However, the Doctor's age fluctuates around 1000 (More on that in another post) but in the Impossible Astronaut, we see The Doctor in his Matt Smith regeneration as both 1103 and 909 (Not to get off track, but this is a huge inconsistency with the 7th doctor's claim of being 953, although of course The Doctor lies, so Who knows(<-- eh? eh? Did you catch that one? :D)?(God, this is a disaster of punctuation.(And yes,I'm putting () in () in ()(Oh, my silly brain!))).

Anyhoo, The fourth Doctor at one point stated that he was 407, meaning that, at the least, each of the four first doctors lived for about 100 years, and (although clearly not true), the first doctor would have at most been able to exist for 400 years. So if the eleventh doctor existed for at least 200 years, but doesn't look like he aged more than a year, how could the first doctor have gone from a baby to an old man in 400 years tops?

3) River ages

River was a little girl when she was playmates with her parents, Rory and Amy, and she is seen to have grown up with them. Of course, River isn't Gallifreyan, so the rules may be different for her (Posing yet another question... is River considered a timelord? Is the ability to regenerate all that is required to be called a timelord, or is there more to it?) Alone, this could just be a little one of Doctor Who's many wonderful little inconsistencies, but it poses a huge question.

Does the Doctor age?

What do you think?