How does sinus infection affect the immune system? Can sinus infection give you coldsores?

Complicated. The reasons that herpes simplex reactivates is not known, but almost any form of stress, whether emotional or physical may cause it in some individuals. Yes, sinus and other infections can result in cold sores, although they are not necessarily a direct cause.
Thanks for asking! Any infection a person has can stress the body's immune system. The more severe the infection the more effect on the immune system. Herpes simplex viruses, which are present in the oral cavity, can cause cold sores with immune compromise or any form of minor oral trauma.
Sinus sores. Cold sores and sinus infections are both often triggered by viral infections. You may be mistaking a viral infection from a sinus infection.

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Chronic fungal sinus infection? If so, then is my immune system bad?

Fungi in the sinuses. Decreased immunity (reflected by deficiencies in immunoglobulins, particularly in iga) is only one cause of unusual infections in common places. Other causes include certain chronic diseases (particularly diabetes, but also rheumatologic conditions like wegener's granulomatosis), various cancers (particularly leukemia), and medications (particularly cortisone). Read more...
Fungal Sinusitis. It depends on what kind of fungus? If it is invasive, then it is likely related to compromised immunity and it can be lethal. Mycetomas or fungus balls are related to chronic allergy (hyper-reactive immune system). Indolent fungal sinusitis, is the exception of being invasive but not immune-related. Only a highly experienced rhinologist can properly diagnosed and treat this condition. Read more...