If a person has a cold (respiratory symptoms) and a flu (vomiting) simultaneously is that a sign of hiv or another immunocompromising illness?

Respiratory and GI : No, definitely not. It is not at all uncommon to have both respiratory and gastrointestinal symptoms simultaneously with a flu-like or other viral infection. This in no way suggests an immune compromised condition.

No: you can get resiratory and GI viral infections at the same time, If you are at risk for HIV then see your doctor and get a blood fest if you are worried.