Darwin's been to the vet. The vet is fairly convinced that the eye issue is conjunctivitis, and Darwin has Chlamydia.
He's been given a metacam jab and I've been told to give him a few drops a day for a few days. I also have to administer Fucithalmic twice a day for around a week. (He said as many times a day as possible, but that's a bit vague for me. If anyone has any advice I'd appreciate it.)
He told me not to worry as it's a common infection in cats. Clearly, I'm still worried, but trying not to go overboard.