Would it be possible to slightly change the trajectory of Ceres to make it end up orbiting the Earth without messing up the Solar System? How much energy would be required and how long would the process take?
It's not possible to "slightly" change it's orbit unless you have a very very long time to wait. When Orbits get in synch with large bodies like Jupiter, for example, then Jupiter can give a tiny tug to Ceres each orbit and if positioned just right, it might nudge Ceres towards the inner solar-system, but only over millions, if not tens of millions of years. That's the slow way.
There's no fast way to move Ceres into an earth orbit without exerting a significant force on it. Orbits are largely stable and to change an orbit that much would requires a serious amount of force.
My math isn't always right, but Ceres orbits the sun at 17.9 KM per S and the earth orbits the sun at about 30 KM/S - now, you might think you need to speed Ceres up, but actually the opposite, it needs to be slowed down to fall into a lower orbit and the amount it needs to be slowed down (My math would be long and clumsy), but the actual deceleration needed is about 1/2 of that velocity difference - so imagine how much energy you'd need to move an object nearly 1,000 KM from side to side, made of ice and rock, twice as dense as water (mass of about a billion billion tons) and you need to accelerate that (decelerate it) oh about 5-6 KM per second (20,000 Kilometers per hour).
Now, a lot of energy could be saved by dropping Ceres to a fly-by near mars and then doing a gravity assist, and then, using earth's gravity as well to help capture it into an orbit around the earth. Mars weighs about 700 times what Ceres weighs, so the effect on Mars would be pretty tiny (all bets are off for Mars' 2 tiny moons though). But at minimum, you still need to decelerate Ceres by about 3 KPS (10-11,000 KPH), and at a billion billion tons that's a ungodly amount of energy, unless you're willing to do it very very slowly, perhaps or by finding a fuel source on Ceres to generate some kind of thrust or moving it so that it's in proper synch with Jupiter so that it gets a small kick towards the sun every so often, (see Mercury/Jupiter resonance below)
I love ideas like this by the way. Ceres would be a nice source of water Though in an earth's orbit it would lose it's water over time to solar wind, so it would be a trade-off moving it closer. Might be better to leave it where it is.
As to your 2nd question, Ceres weighs about 1/80th the mass of the moon. It's not massive enough to "mess up" any planets in the solar system, though if it crashed into the earth for example, it would "mess up" life on the earth big time, but it wouldn't significantly change the earth's orbit. It would change the earth's orbit a little, but only a tiny bit.