Some questions pop up in my mind.
Can the gravitational interaction sling shot an entire stellar system?
How much Km/s2 we can expect these system to accelerate? (In a most
extreme plausible case).
Basically this comes down to gravity assists. A gravity assist is essentially a 3 body interaction. You need a point of reference and 2 velocities. When we send a spacecraft around a planet for a gravity assist, the point of reference is the sun. The maximum gravity assist velocity is (see picture), 2U+v.
I agree with James Kiflinger. Reaching galactic escape velocity is very unlikely because galaxies have large dark matter halos where most of the mass is contained and to actually escape all that requires a very high velocity. The vast vast vast, even with spiral arms extending outwards, still, most of the stars would stay in the gravitaitonal control of the merged galaxies.
From this question 0ur Sun orbits the center of the Milky way at 220 km/s and it's escape velocity 537 km/s so it would need an acceleration of 317 km/s to escape the Milky way. That kind of acceleration is possible given that Andromeda and Milky way will fly towards each other at something like escape velocity speed (537 km/s) - perhaps a bit more as Andromeda is larger, and there's orbital speed (similarish to our sun's 220 km/s), so there's enough velocity to kick a star out if the gravity assist is positioned just right, but you'd need a very heavy star, as a star the mass of our sun simply couldn't effect another star the mass of our sun if they fly past each other at several hundred km/s - the gravitational bend would be much less than 180 degrees, so you'd really need a very large sun, maybe a few dozen solar masses traveling at high speed to (maybe) kick another star out of the combined galaxy. Such an even would be hugely rare but theoretically possible. Now, an individual star could get multiple gravity assists, but 2 stars getting close enough for a good gravity assist would be rare enough, so that, a star actually being ejected, while it could happen, would be very rare. Certainly billions of stars per galaxy it could happen to a significant number of stars, but it would still be a 1 in a million (so to speak) event.
How this acceleration can affect the system? Can it de-stabilise
planet orbits for example?
First things first, accelerations are rare. For a star to accelerate around another star they need to get quite close and in such a scenario, the first thing you'd notice is there would be a 2nd sun in the sky, growing gradually brighter.
Even when Andromeda & the Milky Way collide, form the point of view of your average star, it wouldn't be very interesting unless you actually saw Andromeda's galactic core move towards you - then it might look cool but mostly it would just look like a bunch of stars - like we see in the sky today. The average closest distance between stars in our neck of the woods is maybe about 5-6 light years. In the center of the galaxy, less, but there's still a lot lot lot of space between stars, so even as Andromeda and Milky-way collide into each other, from one stars perspective, most of the time, all the stars would be quite distant and another star passing as close as 1 lightyear would be very rare - like maybe every million years or so, rare.
If our sun was to pass another similar star about 1 light year away, it would be bright in the sky but otherwise the effect would be negligible unless the star was very large.
For a sun like star to have the gravitational effect on Earth that Jupiter has (Jupiter is 4 AU from Earth at closest pass, our sun 1047 times the mass of Jupiter), 32 times further, about 128 AU, which is a teeny/tiny fraction of 1 light year, about 1/470. Even 100 solar masses, such a star would have to pass within 1/47th of a light-year to have the gravitational tug on Earth that Jupiter has. That's crazy-close and would be enormously rare. It would happen to some stars, but your average individual solar-system with life on it would on average not be effected in that way.
Luminosity could be more likely a factor than gravity, as a sun with 100 solar masses would be about a million times as luminous as our sun, and 1 million solar luminosity, to equal the solar output from our sun it would only need to pass within 1,000 AU or 1/60th of a light-year, and a sun that bright would put out lots of UV light. But even so, fly-bys that close would be very rare, but we would still be more likely to be cooked by a large star than orbitally perturbed by one. Smaller stars would need to fly much closer to have an effect, but smaller stars are also much more common.
Outer planets would be more vulnerable and oort cloud objects even more so, to near fly-bys and orbital perturbations, but again, that kind of near fly-by wouldn't happen often.
How it can affect life? Can it affect magnetosphere? change it's
shape? Can solar flares/wind get stronger in some path due particles
As stated, 2 stars would have to get unusually close for there to be any effect at all. The most likely any effect would be interactions between the two stars oort cloud, possibly leading to a big increase in comet collisions, which in and of itself could be pretty devastating when a good sized comet hits a planet. Such events would still be rare, but rather than every few million years, it might be every few thousand (ballpark guess).
Magnetosphere, doubtful. If a star gets close enough to affect a planet's magnetosphere, the planet would be vaporized.
Solar wind/solar flares - also, no measurable effect. A sun's solar flares are internally driven and not changed by objects flying past, especially at the distance of 2 stars even in a close flyby.
If another star passes really near (let's say 1 light year at closest
approach) but at great speed can its light reach that planet at high
energy levels, let's say x-ray and affect that planet life?
Again, no measurable effect. With the colliding of 2 galaxies you might see stars fly by each other at about 1000 km/s but that's nowhere close to enough velocity to create any kind of blue-shifted x-ray effect. Other than the night sky changing more rapidly over centuries than usual, there's no effect to stars flying past each other at 1000 km/s.
Mostly, there would be little to worry about. Star on star dangerously close fly-bys would be rare, even in a galaxy on galaxy collision.