But unfortunately, travelling at light speed (let alone faster) means that you would reach infinite mass and require infinite energy - which is impossible. Atleast thats what Einstein reasoned. Theoretically though, if you could move faster than the speed of light, then you could go back in time.
Speed does have a strong influence on how time progresses, but not ultimately when it comes to things like ageing. If you were travelling at 99% the speed of light for 5 years and then came back down to earth, you'd find that, whereas you have aged 5 years, 200 years have passed on earth. You could take the view that time has moved slower for you because you've aged 5 years whilst 200 years have passed on earth, but I prefer to look at it the other way. If you had a watch on your spaceship, it would only register that 5 years have passed, so according to the way in which we measure time, you have been flying at 99% the speed of light for 5 years and you have aged 5 years - it all balances.
The same applies to our solar system being thrown out of wack, as you suggested dirty. Even though our day would be smaller (23 hours instead of 24), it doesn't mean that 24 hours of events would be compressed into the new 23 hour day - it just means that we'd have to extend the calendar by a couple days to make up for the loss.