Just don't violate ADA on this one. ..don't ask what her disability is, etc.  

Yes.Am I allowed to demand proof, without violating ADA?

We also print our mission statement in our ad as well: "We strive to provide safe, clean, affordable, comfortable and quiet housing for responsible renters in the neighborhoods of west Vancouver."

An ADA violation could turn out to be much more expensive than the damage a pet could contribute. 

When you buy them you can see if they are ADA compliant.

We routinely offer to change out toilets to chair height (also known as ADA height).

You have every right to require her to prove she has a disability, under this act, that requires a service dog.You would be required to allow a service dog, to someone who is disabled under the ADA, if they asked for a reasonable accommodation for their disability. 

Is anyone familiar with the ADA or fair housing act and it would allow us to be covered to allow this house to have all 10 members in the home without an issue.

Instead of spending thousands on an ad a bunch of people are not likely to see have a free try us out day where they get one item free with purchase of another item. 

A service dog is covered under the ADA and you cannot enforce restrictions without violating that.