True, Khan and his crew didn't design the ship, but wherever they got it, why couldn't it have had some armaments? I mean, it was built during wartime, so some country or organization might have had other uses for it, especially if there were enemy colonies or ships near Mars, or Jupiter?
There weren't. We're talking 1997, remember? Even in the Trek universe, there weren't any offworld colonies yet at that point. The Eugenics Wars were a strictly Earthbound conflict, so a vessel designed for extended interplanetary flights could only have been for civilian use. It's not like all civilian activity ceases to exist when there's a war on. The US has been fighting in Afghanistan for a decade now, but in that period we've managed to send a few probes to Mars and the asteroids, with another heading for Pluto.
And did you forget the part where it was a sleeper ship? Everybody onboard was in suspended animation. Who's gonna fire the weapons? Obviously a sleeper ship is not going to be a combat vessel.
It's going to be a retcon, anyway, right? I doubt they would leave the origin of Khan exactly the same, so why should the Botany Bay be any different.
"Going to be?" You're mistaking rumor for fact. We don't actually know who Cumberbatch is playing, but Simon Pegg has overtly stated that it's not Khan, and Karl Urban claimed at one point that it was Gary Mitchell.
One other thing... if they were just trying to get away, why would they have to pick a destination? What if the plan was just to head into deep space, in a random direction, and head back to Earth after years had passed?
I don't think that would be a very wise strategy. They left because they were beaten and outnumbered. Why would that be any different years or decades later?
Besides, it's not consistent with what Khan offered as the reason for their spaceflight: "A new life, a chance to build a world."