Okay so firstly you asked about gravity, so I would say that the planets will not go out of their orbits or either this would affect their orbits. All the planets and matter in our solar system are bonded to Sun through the force of gravity that Sun applies on every matter in our Solar system, Else try understanding Law of Gravitation by Newton too.

It states that:

so, Sun is attracting us and we are also attracting it with the same force

Consider example of a magnet when you pull a metal object with the magnet it will attract it but if you pull your hand fast with a jerk then the object won't be attracted towards the magnet if magnet doesn't have much attraction. Now we are not going like f1 cars through our Galaxy, the process will be slow but fast as compared to other movements in solar system. So, as there would be no hurry, the planets will slowly move throughout the milky way with our Sun.

As you think of Dyson sphere, it can only be used to harvest energy from a star, it wasn't made for protecting a planet, the Dyson sphere will provide kind of more energy and will help to get the energy evenly to the thrusters , What will you do Don't you want other planets in our Solar system? every of them with its own properties. Well if you think to put Dyson sphere on all of them just think how much will it cause us.

Well Shkadov thruster was like moving the Sun without harvesting any energy but it limited our direction to move in the galaxy, I don't think electromagnetism can be used for redirecting the solar winds, they are maybe attracted towards the pole of Earth that's why they enter from the holes at the poles and maybe they made the holes🤔

Of course the Caplan thruster will affect our Sun but the real thing is that it is making our Sun younger, you know one day Sun is going to die but as Caplan thruster will harvest the energy from Sun causing its radiation levels to low down and this will make the Sun younger and will give it some extra life span. where would you bring elements from to feed Caplan? else it is making the Sun younger so its beneficial for our generation.

If we become intergalactic you know what will we do mine the other planets lol, our resources from the planets will like endless we may discover new elements to place them in our periodic table.

however, we aren't advanced at this time but maybe in future we may build the stellar engine.

I hope you have understood my explanation If not, come and talk with me on Discord: IRON MAN#9976