Can a bacterial sinus infection affect asthma?

Maybe/not. The process that triggers a sinus infection can involve other parts of the airways/lung, often viral and this can trigger instability in anyone with reactive airway disease. The sinus infection is usually a secondary event when excessive mucus trapping in the sinuses acquires a bacterial invasion. The processes overlap some but are separate. Some with asthma can flare with any illness

Yes, infection can trigger an asthma attack.

Yes, sinus and respiratory infections can worsen asthma symptoms. One may need to see the PCP for proper treatment of both the infection and asthma.