Can pink eye be spread through your saliva?

Prob not. That's a good question. My answer is probably not. Unless a tear contaminated your saliva and was transferred immediately. My guess is, it wouldn't survive too long in the saliva due to all the enzymes that are there.
Doubt it. But do not know of any data which would support my opinion. Saliva is toxic to many bacteria and they do not survive well in it. Pink eye can be due to both bacterial and viral infections and is usually transmitted by people touching their eyes and then hand to hand to eye transfer. Wonder why you are asking - I guess people still spit on each other.