Tread lightly, my friends. There is lots of conversation about HUD working on this subject, in the meantime, yes you can request confirmation of need of the animal, especially when the disability is not visible.
No, you don't have to accept phony 'certificates' bought on the internet. I suggest that you thank them for bringing the certificate in, tell them you will be happy to keep a copy of it, then explain that your company policy requires a document be filled out by a medical professional who has knowledge of a disability.
Please take care that you aren't punishing a person with a true need for an ESA.
We should be getting some clarity form HUD soon.
Please remember when it is a service animal, you cannot decline due to the breed. The same thing will likely apply to an ESA. Yes, some ESAs are parrots, cats, and other non-canine animals.
And yes, the animal must be on a leash unless the disability requires the animal to be free in order to protect/aid the person. Yes, it is their responsibility to scoop the poop. If they are not physically able they should have someone else do it. Yes, the animal can go where food is served if the person cannot go there without the animal. Yes, it can go in the pool enclosure, but not in the pool.
Be smart, spak to your local attorney and don't get the property sued. its very expensive to make this mistake.