Could tonsil removal help chronic sinusitis and/or asthma?

Maybe/no. Removing tonsils might help with chronic sinusitis if they are chronically infected or enlarged. However, it won't likely help much with asthma.
Adenoids. Children who suffer from chronic sinusitis benefit greatly from removal of the adenoids. Also, chronically inflamed adenoids --> post nasal drip ---> irritation of lower airway. As for as the tonisils go, it would be worthwhile to remove them sooner if they suffer from repeated tonsillar infections and asthma or any other chronic problem that is associated with upper respiratory infections.
No. Removal of one's tonsils will not treat one's asthma or sinus disease. Typically tonsils are removed because of enlargement causing obstructive symptoms or infection. Asthma and chronic sinusitis are chronic inflammatory diseases and one should not these diseases to improve with tonsillectomy.