As an airborne-qualified 35F, can I get stationed in Korea or would I be stationed somewhere else where airborne 35Fs are needed?

Airbone only gives you an additional skill. You will still be a 35F and would typically be slotted as such. Airborne does not really help people get to Korea as far as I know but volunteering with your branch could probably get you here.

Also wondering if I would be doing actual intel work if I go to Korea vs other stations

Your chances of getting Korea are relatively similar either way. Korea is a hotspot for persons, you can take command sponsored tours, deployments, PCs, etc. The problem is going through the right channels. As a 35F your very likely to get the spot. If you are re-enlisting to go airborne however, you will be sent wherever the army needs you to be, this could be an airborne unit or any other unit. Since you are looking to enlist, got to airborne school and then travel out into the army, your chances of Korea are the same as a re-enlistment for an airborne slot, you will be placed based on the needs of the army. But there are still opportunities to travel for a tour in Korea. You will need to link up with the person at your unit that handles deployments, if they do their job, they can walk you through the process. This of course depends on the unit you are sent to and if their scope of duty covers that area, but as I mentioned. Korea is a hot spot right now. So chances are good. As a 35f you can be stationed just about anywhere though, so there is no real garuntee.