It is hard to tell. Though red eyes have the symptoms of redness of eyes, not all eyes appear red means red eyes. But from your description, It is caused by cold not red eyes. In fact, there are many reasons that can causes the redness of eyes. In details:
First, allergy also cause red eyes. If you have use some eye drops, it may cause allergy to your eyes thus make it appear red.
Second, If get hurt to your eyes that will cause bloodshot in the eyes that make it appear red.
Third, People got cold and cough too much can also cause redness of eyes.
That depends. Generally speaking, if you caught red eyes, you will feel itchy and get a foreign body sensation. If it is serious, you may feel that eyelid is heavy and there are symptoms: photophobia lachrymation and a feeling of burning. Red eye could be due to an illness, injury, eye infections, or other condition such as allergies. According to what you said, there are many possibilities: eyestrain, red eye and other eye disease. I suggest you to go to see a doctor at once, for it may cause some more serious problems. Eye disease should not be ignored at any time.
I am not sure. There are some cases that influenza can cause red eyes, but some times, eyes turn to red when you get influenza, but it is not red eyes. Sometimes it is conjunctivitis. What we can do to prevent red eyes is pay attention to clean our hands often, and if you when your have some sign of red eyes, keep away from people and go to hospital.