Without talking about any specific cold medicine, mostly, no. They don't turn off your immune system, they lessen the symptoms of an activated immune system.
As for your second question, you could just as easily turn that sentence around...if someone catches colds easily, they may take a lot of cold medicine...how would you demonstrate cause and effect there?
Of course, I'm sure you're aware that cold medicines don't actually cure colds or get rid of them faster either, they are simply taken to make the symptoms of the illness more tolerable. So, if you have concerns about what they're doing, you don't need to take them and will still get well in the same amount of time.
About the only effect cold medicines will have on actually getting better is that, by reducing the symptoms of the cold (runny or stuffy nose, aches, pains, sneezing, coughing, sputum, etc.) you may get more restful sleep, which will give your body the change to devote its energies to fighting the infection.
Cold medicines usually are symptom-treating rather than pathogenicidal. As far as i know there are no medicine that reduces your aquired or innate immunity to any antigen. In allergy or auto-immune cases, treatment consisst mainly of, as pinpointed before, reduce the effects of an activated immune-system. I.e. antihistamines and other anti-inflamatory.