Can food allergies cause common cold?

Nope. Food allergies can cause a lot of challenges but we can't blame colds on them too. Germs - usually viruses - cause the common cold. Some folks find that dairy makes them more snotty: stuffy nose and ears.
No. But they can trigger allergic rhinitis that shares symptoms of the common cold - nasal congestion, post nasal drip and runny nose.
No. Food allergies do not cause colds. Food allergy symptoms commonly are confined to the GI tract (abdominal cramps, vomiting, diarrhea, bloating), skin (rashes, hives) and sometimes throat or lip swelling.

Related Questions

Can the common cold make your tongue feel swollen? Not food allergy, don't have any and haven't eaten anything yet today.

Possibly. Viral infection can cause hives or swelling and thus tongue swelling can happen. If this lasts for more than 2 days see doctor. If this causes problem with swallowing or breathing, rush to the ER. In the meanwhile, consider taking antihistamine. Sometimes the problem may get worse for about a week before it gets better.