Weepy eyes can be caused by dental problems, although that is more commonly seen when one eye is affected, not both. You can try flushing his eyes with normal saline eye wash for humans. Do you see redness and irritation? In many places eyedrops with fucididine can be bought prescription free. They can be used with rabbits. If flushing the eyes or using antibiotic drops doesn’t take care of the problem, you do need to take him to a vet.
From your description, I think your bun had a choking episode when slime started coming out of his nose. It’s extremely lucky that it ended well. It could have ended very badly. That he pulled his head back was most likely because he was gasping for air. The best thing you can do to prevent future such episodes is to serve his food scattered around his living area instead of in a bowl. Many buns are greedy for food and they can happen to inhale pellets. Any way you can slow his eating down is good. All bun owners should also be aware of how to perform a rabbit “Heimlich”, which is not done the same way as in humans: http://rabbit.org/the-heimlich-maneuver/
After a choking episode, buns can get airway problems, because the mucosa of the airways has been compromized by whatever foreign obejct was aspirated. Bacteria that naturally occur in the airways but are harmless when the mucosa is intact can get a “foot hold” and start causing trouble. It is very possible that your bun’s weepy eyes are due to an airway infection, because everything is connected in there – the eyes, the nose, the mouth and lungs and even the ears via the ear trumpet. An airway infection in a rabbit calls for oral antibiotics, most commonly Baytril. You’ll ned to see a vet to get a prescription. You should try to get treatment as soon as you can, to prevent this from getting worse.