I have a cold right now, but my right eyelid on the top near the inner corner is swelling, red, and is painful. could the two be linked?

Possibly: Maybe in a very nonspecific way. It's probably a meibomian cyst or hordeolum ( plugged oil gland or a sty). Try hot compresses. Increase the intake of flaxseed oil in your diet..It will thin secretions.

Stye: You have a stye of your upper eyelid which is an infection of a hair follicle caused by contamination from the cold from your nose use warm compresses so the stye can come to a head and drain see your doctor for an antibiotic eyevsalve or eye drops.

It is possible: Your eyelid pain and swelling could be anything from an early stye or chalazion to a lacrimal drainage (tear drain) infection, to orbital cellulitis. Or it could be something more significant. You should see an oculoplastic surgeon or an ophthalmologist right away.