In over 50 years of Doctor Who, there have been many mysteries that have come up. Some, like "What planet is the Doctor from? What species is he? Is Adric the most annoying companion ever?" were answered a very long time ago. Others still linger. Now while all of these mysteries are ones that haven't yet been answered, many of them are also ones that should never be answered! We don't need to have every little question spelled out for us. It's good to have some mystery. It's unlikely that anyone really wants to know the name of the Doctor, or expects that we will, at this point, actually get an origin story for Mel. On the other hand, perhaps we will, some day, get the full story of why the Doctor left Gallifrey, and why he took a dangerous super-weapon with him. So bear in mind that this isn't a list of mysteries that should be solved, or things the show tells us. It's just a list of mysteries that haven't yet been solved.
7. Why Did The Doctor Leave Gallifrey?
We've had vague hints over the years. We know he was unhappy living on Gallifrey, and that he had an itch to go explore. But there must be more to it than that, surely. When we see him and Susan leaving the planet, it almost seems like they're doing it just a step ahead of someone who is after them. Heck, when we see them in "An Unearthly Child", the Doctor seems paranoid and fearful. We know from "The War Games" that he was absolutely terrified about contacting his own people, seeming to imply that he was wanted for a crime of some sort. Of course we did later find out that the Doctor may have been on the run for stealing a super-weapon. He was carrying something called The Hand of Omega, and it's the sort of thing that can apparently make stars go all explodey. Very early on, he traveled to Earth and hid it there in order to prevent it from being found by some alien force. That force was later discovered to be the Daleks, which the Doctor hadn't met prior to leaving Gallifrey, so that creates a bit of a problem. We can assume he knew someone was trying to steal The Hand of Omega, but not who. Or it's possible there was a power-struggle on Gallifrey, and the Doctor ended up on the wrong end of it. With the various political plays the Time Lords seemed to endlessly indulge in, this is entirely possible. It's easy to imagine that in this case, the Doctor's life may not have been at risk, but perhaps pretty much everything else he had was at risk. So leaving Gallifrey makes some sense. It might also explain why the only charges against him the Time Lords actually listed in "The War Games" were his persistent meddling with various other cultures and histories.