It sounds like a cold. I used to use children's cough medicine in small doses and i seemed to work. Best bet is go to the vet and let him have a good look.
It looks like that your cat has allergies, In order to avoid all sort of dust allergies you should vacuum more often and avoid outdoor allergens, it is better to have a air purifier in your home. You cat seems to have purulent nasal discharges in which it seems that it has fungal or bacterial infection.
I think you should get your kitty to the vet, discharge from nose for longer time is not a good sign, a vet can easily clear the nose and may also suggest you some meds as well.