“Can an eye infection last for months?”

If you do have an eye infection, the classic signs of most infections will include redness, tenderness, swelling, pain, eye discharge, and possible blurry vision.
I would strongly suggest that you see an eye doctor ASAP in order to determine exactly what your problem is. Any infection that "lasts for months" is an immediate red flag. Treatment sooner than later is always the best option and to not wait until this can become worse and may take longer to treat.
Another question that I have for you is related to what part of your eye do you think is infected? Something like a conjunctivitis can be a viral, allergic, or bacterial problem and they are all treated differently. If you have a corneal infection that is bacterial, this would need to be looked at by an eye doctor in order to prescribe an antibiotic drop to treat it. And if you have an internal eye infection, this would be something serious that would need to be seen, again, ASAP by an eye doctor. If you have had or still currently have any blurry vision,
please call an eye doctor. Good luck and best wishes.

Yes

yes, If you do not treat the infection it can change to the chronic form and stay longer.

If it’s recurrent , then yes

Yes! Depending on what is going on, certain infections can last a long time if not treated properly. If you are having any other symptoms (difficulty or pain while urinating) I recommend seeing your optometrist to rule out bacterial infection.