It'll have basically no effect on the planet's orbits, except perhaps on the theoretical planet 9 and even then, not very much. As close as it's expected to get Gilese will till be too far away to affect the 8 planets much.
Gilese 710's mass is about 60% the mass of our Sun.
I've seen two estimates, of 13,000 Astronomical Units (AU) and 13,365 or about 77 light days or 0.21 light years at closest pass, and some estimates put a +/- margin of error at as much as 20% of that, but lets use 13,365 AU as an estimate. For a star, that's exceptionally close but it's enormously distant for a planet.
Neptune, the most distant planet, averages about 30.1 AU from the Sun, so Gilese will be about 440 times further away from Neptune than Neptune is from the Sun. That will give it a negligible gravitational effect of less than 1 part in 300,000 on Neptune's orbit. That's, by crude calculation, a bit less than the effect Saturn has on Earth's orbit. That's enough for a small perturbation, but you'd need some pretty high tech equipment to even detect it. It's about 50 times less than the gravitational effect Neptune had on Uranus when it was noticed that Uranus' orbit was behaving oddly and that observation lead to the discovery of the planet Neptune.
So, basically, for the 8 known planets, the outer planets will get small pushes, noticeable by very meticulous observation only and the inner planets, essentially no effect.
It's effect on Planet 9 would be greater but still small. The estimate for planet 9 at Aphelion is about 1,200 AU or 1/11th the closest pass of Gilese. That's close enough for a noticable push. Not enough for any major changes like pushing it's eccentricity to where it passes anywhere close to Neptune, or ejecting it from the solar system, but Planet 9 could be jostled somewhat, perhaps as much as a 1% change to it's orbital period and/or eccentricity, which for a planet's orbit, 1% is a large change over a single orbital period.
The bigger effect, as you mentioned, will be on more distant objects, several thousands to tens of thousands of AU from the Sun, or several times more distant than the theoretical Planet 9 at it's most distant. Objects that distant have never been observed, but are theorized to be out there in vast numbers based on a handful of observed objects with very high eccentricity. Oort cloud objects at those distances will get cast every which way, some cast out of the solar-system and a small percentage, pushed to higher elliptical orbits where they could pass through the inner solar system. Our sun will do the same thing to Gilese 710, casting it's Oort cloud objects in every direction and each star will steal a number of orbiting objects from the other.
The initial effect, the inner solar system might pass through the outer edge of Gilese 710's outer oort cloud. I don't think anyone knows how extensive it's Oort cloud is at 13,000 AU, but any initial increase in objects passing through the inner solar system will be from Gilese 710's Oort cloud.
Any of our Solar system's Oort cloud objects that Gilese 710 sends towards the inner solar system will take (roughly) between 50,000 and 400,000 years to reach the inner solar-system, so Gilese 710 will be perhaps still visible to the naked eye when the proposed rain of comets begins, but it will be long past being the brightest star in the sky if/when any objects it sends towards the inner solar system make it here. The reason for this is that it moves much faster than any objects it's likely to send our way. Gilese 710 will pass through the solar system at about 50,000 kph and any objects it sends towards the inner solar system, because they're so spread out and most are likely to just get a small push from the passing star, they will leisurely drift towards the inner solar system at roughly between 500 and 1,500 kph, requiring about 50,000 years for the first ones to begin to reach the inner solar system and become comets or perhaps crash into the Earth or Moon or other planets.
Also, fun fact. At closest pass, Gilese 710 should be a hair brighter than Jupiter at Jupiter's brightest, nearly -2.8 apparent magnitude. It will be the brightest star in the sky for ballpark 15,000 - 20,000 years, over that time-frame it will move across the fixed stars a good percentage of the 180 degrees of horizon. In a human lifetime it might move a bit over 1 degree of arc at closest pass, so it won't be very noticeable that it's moving, but that's very rapid proper motion for a star.
I've excluded the mathematical formulas cause I don't do the that well and it would get a little ugly. I can't do the math of a 3 body system, so this is an approximation, but I my approximation should be in the ballpark.