Is it obligatory to inquire about the content of the perfume sprayed in the mosque before the Eid prayer, which one fears may contain alcohol, or should one act upon certainty (of purity), especially since alcohol is volatile and may not have fallen on the ground?

We advise you to turn away from whisperings (waswas) and not pay attention to them, as you are afflicted with a great deal of them. You may adopt the easiest opinions until Allah Almighty grants you recovery. This is not considered blameworthy pursuit of dispensations. As long as you are not certain that something contains alcohol, the default is its purity. Even if it is assumed to contain alcohol, as long as you are not certain that it fell on the ground, the default is the purity of the entire place. You have latitude to adopt the opinion of those who consider alcohol pure.