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Protrusion of one eye is commonly seen in hamsters. The hamster needs to be seen as soon as possible by a vet familiar with their care for therapy. Until you can get him/her in to be seen below are some things you can do at home. There are several possible causes of this sign in hamsters. The most common is a cheek pouch abscess (or abscess behind the eye) that is pushing the eye outwards. This is usually caused by a seed lodged in the cheek pouch or other food item that may no longer be present, or is still present (impacted cheek pouch) and causing secondary infection. Other problems that can cause this include trauma to the periorbital area or inappropriate/rough handling (skin at back of neck held too tightly when animal is restrained). Some hamsters can develop dry eye (keratoconjunctivitis sicca) and secondarily the eye protrudes outwards.
At home, cleanse the eye gently with a sterile eye wash that you can purchase over-the-counter at any pharmacy. Then, lubricate the eye well with artificial tear lubricant (not just a liquidy solution but something like Celluvisc). If the eye is truly popping out of the orbit, a veterinarian will need to replace it. Medication is often needed to help clear any infection and/or treat dry eye issues. Rarely, the lids need to be sutured together to hold the eye in place and allow the muscles that support the eye to heal. Sometimes if the eye has been proptosed (out of socket) for too long, we cannot save the eye and enucleation is a very viable alternative.
Best of luck.
Probably the best way to handle a hamster that likes to bite is to grab and gently wrap in a washcloth. Helps you have a bit more confidence that it might bite the cloth and not your hand. Of course, gentle scruffing of the loose skin of the neck is acceptable, but you are right that this can certainly cause a bulging eye to become worse.
I would also be suspicious of a tumor that is behind the eye. This could affect the nasal cavity (it could be growing up from the nasal cavity and pushing her eye outward). This would cause small capillaries that conduct blood through the nose to break and cause the bleeding you have seen. We still cannot rule out an abscess (be it from a tooth root or trauma from a seed or other food item) since these little guys like to form abscesses easily. Therefore, I recommend you do your best to get the antibiotics into her. They are tough to medicate when not having nasal/ocular issues. If the antibiotics don't help and she continues to decline (stops eating, losing weight, difficulty breathing worsens, very antisocial) then it would probably be the kindest thing to put her down. I do hope the antibiotics help and you don't have to make this decision.
Regarding pain, there is absolutely medication you can administer and I would recommend you ask the vet you visited. At a minimum (s)he can dispense metacam (meloxicam) which comes in a suspension and can be used in these little guys once daily for pain relief.
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