First, that is awesome you rescued this little guy that has multiple issues. Multiple issues = harder to adopt. So great on you!
I vote for go see the vet, quick quick quick. When I am in clinic, the receptionists will come ask us as we are buzzing between exam rooms about clients who call in - should they wait n’ see, should they book an appointment, or try this or that or polysporin.
For many things you have some time to sort it out, but for anything to do with eyes, you gotta get it checked out quick. That is always my answer for eyes. Because when the eye goes bad, the eye tends to go real bad. Kind of like child actors.
Underneath the raised third eyelid could be viral keratitis, a corneal ulcer, a symblepharon (adhesion of structures in the eye from inflammation) or other situations, ranging from simple to complicated. Poor guy! The asthma and hyperesthesia are probably innocent here - it's the virus causing the repeat upper respiratory infections that may be responsible. Unless he's an outdoor kitty that got into a scrap. I'm all for home remedies when they work, but I would also vote for not putting anything, other that saline, in that eye until I knew what was going on. And I would want to know what was going on ASAP. Which is why I tell the receptionists get the person who is on the phone in, even if it’s over the “dinner break”. Which is most days reduced to “snack break”. Which on many days is reduced to “bathroom break, wash hands and shovel spoon of peanut butter in mouth before next walk-in appointment”.
The upper respiratory tract viruses (which is my first assumption of what is going on here) are stubborn. They require the patience of a saint and multiple kicks at the can to deal with. L-lysine is something to ask your vet about as well (an antiviral). Talk to your vet about doxycycline. Also talk about finding which eye medication will work (often a trial and error process, where some seem useless and some become priceless). Many of these guys get better with time if it’s one of the nicer viruses (which is why i might have them tested for the bad viruses if the budget permitted…).