Does a cyst 6mm in diameter in the left horn ventricle cause chronic migraines?

Arachnoid cyst. I don't think so, this cyst is small and harmless ausually discovered accidentally when head ct scan or brain MRI is done as a work up for headache.
Unlikely. According to your short description, probably not, but you should have a discussion with your doctor and a thorough review of whatever imaging study you had done to ensure proper treatment or follow up is performed.
Unlikely. 6mm choroidal cyst would not obstruct a normal sized adut ventricle and would likely be incidental and asymptomatic.