The dishes definitely need cleaned out daily and more if you notice fecal matter in or on the dishes. It is a really good idea to have a second set of dishes so you can just switch them. Then you can clean the first while they have the other set in and then switch back again.
As far as the eye problem this matter really needs a vet. I really doubt cleaning the cage will "cure" this (although papers should be changed every 1-2 days and cage should be scrubbed around once a week.). It sounds like an infection or maybe a serious allergy and there is nothing over the counter to treat this. A vet would probably give you an antibiotic depending on what the cause of the infection is. Depending on what you are using in the bottom of the cage there is a remote possibility that she has gotten somthing in he eye that has scratched it causing an irritation. If this is the case gently cleaning the area around the eye and fully cleaning her cage should make the irritation go away but I would still suggest a vet if she has never seen one.
As far as adding other birds to the cage. First only after her irritation or infection has cleared up and then the cage is not really large that you described. You might be able to add one more budgie if they get along (either way you would need a separate cage to begin with for quarantine and them getting used to each other). The only way I would suggest adding another bird though is if you can quarantine them, then introduce them slowly and lastly are able to allow them out of the cage time or else the cage would be too small!