Lots of discussion about terraforming the moon involves smacking it with a comet such that it rotates with a 24 hour ish period. But I don’t know if this will affect its orbit in any way. Is it possible that imparting a faster spin to the moon would affect its apogee and perigee?
A change in the rotation of the moon, or even a change its mass, will have no effect on its orbital parameters.
That said, how you would change these things probably would change its orbit in small ways. Colliding comets into it will impart a small force which would change its orbit by a tiny amount. The type of change will depend on the angle of the collision; in addition to changing the apogee and perigee, you may see inclination and eccentricity changes.
If you have control over the timing and the angles of the impacts, you could arrange it so that the changes to the orbit are cancelled out, while still keeping the rotation.
I think the critical question is: Exactly how do you "spin up" the Moon? You can't apply force externally to an object in space without imparting both spin and linear motion, but, like spinning a basketball on your finger, the closer to imparting force tangent to the sphere surface, the closer you come. Or, you apply to equal and opposite forces at opposite ends of a diameter. That will create pure rotational motion.
And so long as you don't disturb the Moon's orbital velocity (or not very much :-) ), the moon's rotation won't affect the gravitational force it applies to the Earth.
And now someone is correctly going to point out that the Moon is not homogeneous so if it spins, the CM will change relative to the Earth, leading to some orbital oscillation, etc etc.
For a single impact to accelerate the moon's rotation to once per 24 hours, the impactor would have to be much larger than a comet. Imagine suspending a stone ball with piano wire and trying to get it spinning by throwing tiny pebbles at it. Even if you cover everything with velcro so the pebbles don't just bounce off, you're going to need a sizeable rock to get the moon spinning at a decent speed. I haven't done the math, but I would expect this kind of impact to melt the moon's crust and probably accelerate enough debris out of the moon's gravity well so that people on earth would be endangered.
An impact like this will impart linear velocity, which will change the orbit some. The amount will depend on the angle and velocity of the impact and the mass of the impactor. But I have the sense that the change might not be very large. Whenever I have read about the giant impact thought to have created the moon, I haven't seen anything about how the earth's orbit around the sun would have been affected.
You could spin up the moon by lots of small impacts arranged to allow heat to dissipate and to balance out the direction of the force. Then there would be no immediate change in the orbit. However, over the next few billion years, the moon would move away from the earth at a faster rate than it is now, due to tidal effects, with its rotation gradually slowing over this long period.
I feel like the real question is, what would happen if the moon were spinning faster, and everyone is answering the question, how can we make the moon spin faster?
I assume you know the mechanism. Back in the past, before the moon was locked, the earth and the moon were closer together. The moon slowly spun down and became tidally locked to the earth, and, since angular momentum is conserved, the distance between the earth and the moon increased to preserve the total angular momentum of the earth-moon system.
The moon is tidally locked with the Earth, but the Earth is not tidally locked with the moon. But the moon is trying to lock it! The earth is slowly spinning down, losing angular momentum, and the moon is slowly moving outward, about 3.8 cm/year, preserving the total angular momentum of the earth-moon system.
So it matters that the earth is spinning! When the earth becomes tidally locked to the moon, the average distance between them will be constant (elliptical orbit). What if the moon were suddenly made to spin? It would be like the earlier version of the earth-moon system again -- the moon would slowly spin down and the orbit would adjust to preserve the total angular momentum of the earth-moon system. The distance between the two bodies would be changing faster than it is now. (Or, if the moon were made to spin the "wrong" way, and fast enough, they would be separating more slowly or even getting closer.)
That's kind of a heuristic analysis based on the knowledge that the bodies will tend toward a tidal lock, and that angular momentum is conserved. As for practical effects, if we wave our hands and say twice as fast as now, or three or four times as fast, it's still a very slow process, and your children's children won't notice a difference. (I'm sure a definite answer could be given, since the question specified a 24 hour period, but I don't know how to calculate it.) So, other people have already said "no effect", and that's basically correct.