It does, to some degree, but ultimately those *symptoms* are mostly how your body responds to threats, not the effects of the cold itself.

The bug doesn't give you fever, diarrhea, and a headache. Your body responds to the bug by inflaming the infection (more liquid/antibodies/resources for the immune system) which causes diarrhea. Your body increases the temperature because lots of bugs don't like the heat. This is your fever. Your body could increase blood pressure, too, which could result in a pressure headache, for example.

And because its your body responding, it's going to respond very similarly each time. You might always get super congested, but my most common symptom is a headache.