Might it be possible that a a sinus infection cause a stroke?

Unlikely. Unless the sinus infection becomes invasive and affects the vessels or erodes the skull, it cannot cause stroke. The chances of a sinus infection becoming invasive are very rare.
Very unlikely. However, if the sinus infection is not properly treated and invades the base of the brain, it can cause the cavernous sinus (veins) to clot, which can cause a venous stroke. Some fungal infections of the sinuses can invade/destroy tissues at the base of the brain as well.
Unlikely. Most sinus infections are caused by bacteria, they are usually limited to the sinuses but sometimes can spread and cause meningitis. Some rare cases are caused by fungus which can erode into the brain, cause clots in the blood vessels of the brain but not an actual stroke. You can have both, but that does not mean one caused the other.