During the Civil War would a soldier ever belong to a regiment from a different state?

Sometimes, especially if he was an exceptional officer, he might be transferred to another regiment that needed one, regardless of where he (and they) were from. But, you're right to be suspicious. Most of the regiments were raised from a county or two; most of the soldiers fought (and died) next to their neighbors, cousins and friends.

Men who felt strongly about slavery (or property rights, as the Southerners put it) would sometimes switch; a pro-slavery man from Ohio would fight for Virginia, and vice versa.

If the name is John Smith, be really careful. If it is Ralph Kablonski, you can be almost certain you've got the man.

Look at how many men with his name and born in the year he was born (+/-) 3 were in Kentucky in 1860, and how many in Virginia.

Finally, a lot of Virginians moved to Kentucky to get away from the crowds; when 1861 rolled around they enlisted in local regiments.

It is possible, especially if he fought on the side of the Union. Many who remained loyal to the Republic fled Virginia in the early parts of the war and some joined regiments in neighboring states.

You might want to check where his family lived at the outset of the war and where they settled after...and if you had any more distant relatives in Kentucky. If they lived someplace like Charleston or Huntington (remember that modern-day West Virginia was still part of Virginia then) and sought to relocate, it would have been likely for them to relocate somewhere that they could be with relatives.

Yes. AFAIK, people could enlist in a regiment from any state they wanted. Most regiments were organized from a particular region and so people tended to be from that state, but they didn't have to be. I think that cross-state enlistments may have been particularly common for Southerners looking to fight in the Union army because they couldn't always enlist in a regiment formed in their own state.