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Welcome to Just Answer and thanks for your question. It think it's just a matter of terminology. A "cold" is a viral upper respiratory infection. Your body's immune system fights the virus and in the process produces lots of mucus, and pus which you blow from your nose and cough up. There can be viral infections of the eye; we call it viral conjunctivitis. The eye is usually red and swollen and watery. What you are probably experiencing is your eye's normal function of cleaning itself. At night the tears puddle in the corners of the eyes and can become crusty. Any upper respiratory infection or eye irritation or allergy will also cause an increase in tears with additional puddling and crustiness.
Yes, this can be the result of a slight viral infection, or of an allergy. Either will produce bloodshot eyes. Warm compresses are fine as often as you would like to use them. Viral infections usually require no medical treatment and get well on their own in just a few days. If the eye feels "scratchy" or irritated, artificial tears from the drugstore will also help.