Also whether you rent or own
I'm not sure with how long adoption takes assuming it depends on the pet you choose. Also it will be up to date on all shots and spayed and neutered before you take it home or you will have to set up an appointment usually within a week to have that done.
I waited a full 2 weeks for my pet. I'd say no, it'll take at least a little while for the pet. And no, they don't check homes, but they check some documents like... proof of ownership, etc.
Not to sure about the adoption part the animals I have were either found on a street or from my family.hopefully someone else can shine some lite on ur question.
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