He's supposedly half human on his mother's side, which would indicate she was likely to have given birth to him during Earth's recorded history. When was The Doctor (from Doctor Who) born?
As with all Doctor Who questions, the canon answer is "who really knows?".
Aside from the fact that we can't trust anything The Doctor actually says, there have been at least three completely contradictory explanations for where he "came from" over the course of the show. Which one you consider canon is mostly a matter of which one you like better, though my impression is that most Whovians go with the first one.
Born On Gallifrey
Based solely on his dialog on the TV show, The Doctor is a full-blooded Gallifreyan that was born some unknown number of centuries ago, on Gallifrey. Word of God (TVTROPES WARNING) say that we've actually seen his mother on-screen -- the elderly woman that appears to Wilfred, and with Rassilon, in The End of Time was intended to be his mother, though any on-screen confirmation was removed. (The closest we get was Wilfred asking about her and The Doctor pointedly not answering.)
Based on this, we have no idea when he was born, but we can get a good estimate if we ignore a few outlying details (like the way they judge figure skating). Comments by The Doctor, Romana, and even the human-form Tardis put the doctor at something around 900 years as of the last season of the program, and around 200 when he first stole a Tardis and went travelling. He grew up and spent his teenage years on Gallifrey, starting Time Lord training at age of eight. We've even seen his Gallifreyan cradle from when he was an infant.
We have no idea what year it was on Gallifrey when he left the planet, or even if Gallifrey uses our limited, human understanding of time. (Even humans these days know that "time" is not a fixed quantity, but is measured relative to the observer.) Either way, according solely to the television-show-canon, he was not born on Earth at any point during Earth's recorded history. The first time we know him to be on Earth was in the 1950's as The First Doctor, though he was on Earth at least as early as the destruction of Pompeii as The Tenth.
Exactly once, in The Doctor Who movie, the Eighth Doctor makes a comment that he is half-human on his mother's side. Unfortunately, this becomes a crucial plot point in the movie; he only managed to "save the day" because this statement is true. So we cannot just write that line off as typical Doctor deception.
Other than the movie, however, there is absolutely no evidence that it's true, and plenty of on-screen evidence that it's false. (There's also no evidence that The Master is half-lizard, for that matter.) Most fans just tend to write off any contradictions from the movie as non-canon and ignore them.
If you read the last few Seventh Doctor's New Adventures books; particularly Lungbarrow and Cat's Cradle: Time's Crucible, you learn that Time Lords aren't born at all. They are infertile, due to some ancient cataclysm, and Rassilon set up a series of Looms that literally weave together new Time Lords fully-formed. According to this theory, The Doctor doesn't even have a mother. Lungbarrow claims that The Doctor abandoned his House and his cousins about 700 years earlier, which more or less matches the time frame that The Seventh Doctor would have been on-screen.
Unfortunately, this contradicts a ton of on-screen references, particularly any time The Doctor talked about having or being a child. According to the TARDIS Index file entry on Looms, there is an Eight Doctor Adventure (which I've never read) where he addresses this inconsistency by saying that he remembers being loomed as well as being born, but knows that one of the two is just a dream.
We don't know. We know what he says his age at various points (this wiki page has a good list, complete with references to the episodes they were mentioned in) but, as Steven Moffatt (the current Doctor Who showrunner) pointed out:
The thing I keep banging on about is that he doesn't know what age he is. He's lying. How could he know, unless he's marking it on a wall? He could be 8,000 years old, he could be a million. He has no clue. The calendar will give him no clues.
We also don't know that his mother gave birth to him in Earth's history, given she would have been involved with the Time Lords and hence may well have given birth to him on Gallifrey or on some other world altogether. It may have been before humans walked on two legs, or after the Earth's destruction as seen in the new series. It has never been revealed in the series.
For the time lords, all realities exist as long as they don't create a paradox. Which reality the Doctor is referring to depends on which reality he's perceiving at the moment, much in the same vein as Schrodinger's cat. However, time is mutable for a time lord, so unlike most other beings, the simultaneous realities of the Doctor's past snap back once the moment of perception passes; i.e. he stops thinking about it. This allows for different versions of his history to be true at the same time. It's also why he makes seemingly contradictory statements or at times appears not to know which answer to give. He's trying to translate his perception of time and reality to people who view time as linear and reality as a singular event. As a side note, the paradox factor keeps the number of realities in check so that they're reduced to a manageable size, instead of being infinite and beyond the ability of a finite being to comprehend (though because of their vast intelligence, time lords are able to comfortably exist within the framework of multiple realities, which is a feat only one human being has ever accomplished; that is, of course, Clara Oswald). Boo-ya!