Our own galaxy is on a collision course with Andromeda. Will it hurt?
No, the stars (and planets orbiting) residing in each galaxy are too far from each other to collide or really interact in any way. Perhaps in the inner regions of the galaxies (around the central nucleus) some may disturb one another to a great extent, but out were Sol resides between spiral arms there is no danger from collisions. Only the interstellar dust and gas from both galaxies would collide, resulting in some spectacular light-works or even nova. This would take place over thousands of years.
Define "hurt". The galaxies themselves will be shredded by gravitational interactions before recoalescing into an elliptical, but the individual stars and planets will mostly survive intact. There will be a huge burst of star formation that will increase the chances of being in close proximity when supernovae take place, and some star systems may be ejected into intergalactic space (Which would be a lousy place for a civilization to find itself, bad enough when stars are "just" a few light-years apart, try interstellar travel when the nearest star is hundreds of light-years away.), some might get thrown into the central supermassive black holes at the center of each galaxy. Infalling material will trigger a quasar outburst from the central sbh which would fry any systems in the path of the jets. As far as Humanity and Earth goes, the Sun will be its death throes by then so it's a moot point anyway. Either we have the technology to avoid danger or we'll be long, long extinct.
Recent data suggests that the Milky way is more massive than Andromeda, which previously(to my knowledge) it had been Milky way was larger while Andromeda was more massive. Collisions between stars are extremely unlikely, though are possible . And obviously the super-massive black-holes will eventually collide. Some simulations show we will become an elliptical galaxy whiles others show we will remain a disk galaxy with new arms and trails as some systems are thrown either out of the galaxy system(s) or thrown on extreme elliptical orbits extending out towards the equivalent of a galactic "ort cloud" before beginning their journey back.
I think you have the basic mechanisms correct. The way the natural systems play out. This is a science fiction and futurism forum so there is no need for natural.
The collision with Andromeda is likely to be slightly before the red giant branch phase, 4 billion vs 5 billion years. Earth would depopulate in around 1.1 billion years if there is no sunshade or moving of the planet.
The red giant branch (RGB) and asymptotic giant branch (AGB) phases of the Sun will be its most energetic. The scale of the Dyson Sphere will increase by 3 orders of magnitude. The RGB stage will last over a billion years. AGB stars eject large parts of their mass naturally. Starlifting becomes much easier. Some people like the idea of starlifting before entering the RGB. In that case the timing would change. If the Sun should eject around 0.4 to 0.5 solar masses of material.
This is not a bad thing for a civilization trying to colonize other galaxies. For a K0 civilization like us now it would be fatal. Kind of like the frog launched by NASA in 2013. For an organized K3 it is an ideal propulsion system. You guide stars through a gravitational keyhole where the tidal force is lower than the surface gravity.
I assume by "system" you mean a planet in solar system like Earth is. The jets are ionized material so they should respond to magnetic field. Of course Earth and the Sun's magnetic field would be overwhelmed a long distance from an active galactic nucleus. We can do much stronger magnetic fields. Allowing a huge supply of energy and momentum to fly into deep space unused would seam wasteful.
A K3 civilization could avoid that. The big collision cannot be avoided but passing stars can kick each other towards assigned destinations. If you line stars up for direct impact you can change course by shifting over 1 solar radii. I am not sure if elliptical are a good or bad development though.
Either way it is an opportunity to rearrange everything. The Dyson spheres around stars or free flying swarms can ram scoop the interstellar medium. Loose stars can be gathered into clusters. Clusters that are becoming too dense can use passing stars to energize their cores. Binary system can use passing stars to boost orbital distance.
High velocity direct collisions almost never happen naturally outside of dense clusters. Astronomers disregard that possibility and assume all mergers spiral down. If we can aim the stars all options are open. Imagine a white dwarf star on a hyperbolic orbit that passing through the radiative zone of the Sun. The white dwarf should helium flash on the way through.
I have uncovered some new information on this event that might interest all of you. According to new calculations based on data pulled by the famous GAIA satellite, the Milky Way/Andromeda collision will occur in 'only' 3.9 billion years, instead of the long-touted 4.5 billion years. You can read more about this newest revelation HERE.