Who determines if a soldier receives campaign credit for his or her service? Is it the Unit commander or higher?

Your orders to the designated Area of Operations and possibly a manifest. No Commander's signatures etc. your S1 should do this as part of redeployment.

You receive a campaign medal per your orders for deployment to that theatre of operations. It has nothing to do with command at all. The orders come down to say deploy to Southwest Asia as part of operation X. You should receive the requisite campaign credit whether you are there an hour, or the whole duration. It should be reflected upon you DD-214 automatically but if it is not then simply see your units personnel section to have your 214 amended to reflect it. Just make sure you have your orders.

When you deploy you will receive something called a Statement of Wartime Service that would indicate that you are authorized specified awards. When you demobilize those will be included in your DD214.

It becomes muddled a little if you are deployed to one location (ex. Kuwait) but you get TCS orders for Afghanistan, then Iraq, syria, etc. depending on how long you are in each location you may be eligible for other awards, usually have to be 30 days or more in each location. If that is the case check in with your admin people to get clarification regarding awards then, but my understanding is in those cases you will get the appropriate awards associated with that location.