If I close eyes real tightly I get a buzzing sound in one ear for 5 or so seconds. What is this?

Tinnitus. Squinting hard tenses the facial skin, slightly pulling the external auditory canal and outer layer of the eardrum, which may temporarily amplify tinnitus. It's normal. Take care!
Sure. There is a strange connection between your throat and ears that you may be aware of "popping your ears" or "swallowing on a plane". oddly enough squeezing your eyes closed may engage a muscle in this area. This alone changes the sound in your ears. Your carotid artery is just behind your ear drums and if you hear a rhythmic pulsing it is that. If this is your only problem, nothing scary here.