Best recommendation I have, connect with a good REaltor in your area of interest.... when renting
pets are usually an issue, and your Realtor will be able to explore the options and terms beforehand, so that you will not see and fall in love with a rental unit, that either does not allow more than 2 pets, or no dogs, or no cats etc. etc. every buildings rules are different....
In your situation you do need a Realtor who likes to work with renters and will get the info ahead of time to show you only what will work for you, and outdoor space will be more difficult yet.
Good luck to you...
Edith YourRealtor4Life! Working always in the very BEST interest of her clients
with @Properties covering the city of Chicago, N and NW suburbs of N. Illinois, and the
fine homes on the North Shore - and with partner agents all of the US and properties worldwide