If you put spin on an object in motion in space will its path curve?

It will conserve angular momentum and spin, and it will conserve linear momentum and travel straight. It will not curve if it’s not acted on by an external force.

@Caravanfan is correct because curve balls need air + their spin to generate the force that makes them curve: http://mechstuff.com/why-a-spinning-ball-follows-a-curve-path-during-flight/

Correct. This was proven when Alan Shepard hit a golf ball on the moon.. It flew straight, without slicing or hooking.

Newton’s first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force.

The curve of a football is due to aerodynamics.
No such thing in a vacuum, so it will fly in a curve because it is still affected by gravity.
Speaking of which, there is no such thing as “zero gravity”, because no matter how great the distance, there will always be gravity, albeit miniscule.
The correct term is “microgravity”.

No. All forces are the same in all directions.

What if it’s not spinning around it’s center of gravity?

It would still tend to go in a straight line. It might be a crooked, curvy line, but a straight line none-the-less.

Now I’m wondering if it can spin around anything but it’s center of gravity. I’m picturing getting it spinning while captive, and then throwing it. In my mind it can’t keep spinning eccentrically.