My question is should I just wait to make an offer once the tenant finds a new place, or should I go ahead and put an offer in with an addendum stating that the tenant will not renew the lease?

@Kevin Hill , I'm seeing this in my market, too.While I have no sympathy for the banks that lost money during the big foreclosure rush a few years ago, I also cannot fault them for wanting to put $85k into a property and make a boatload. 

Alternatively if you feel the seller may not agree to pay your brokers commission, you might be ahead if you get your broker to agree to a fee that you will pay if the offer is accepted.A real estate agent can't solicit a listing that is already being offered by someone else.

Go ahead and list it when it's ready.

He's gonna sell to someone.. could be a consideration if price was right and your prepared ahead of time to have the cat lady gone. ya if you do make offer I'd not accept pets in future. 

If I go ahead with finding/purchasing a duplex in the next few months, will it affect my chances of approval for the car loan by the end of the year?

Ask questions (write these down too if you need) such as:-Why are you moving-Where did you live before-What did you like about it there-Where do you work/What do you do for a living Then if you feel good about it after that go ahead and tell them about your offering and make sure you preface your requirements with a sentence like: "Great, now here are the exact requirements to rent this unit...blah blah blah blah" Right down your list.Once you finish, be certain to ask "Do you meet all of those criteria?"

A service contract may be the way to go.You let your tenants know ahead of time your coming out for inspection of unit and you maintain your building, the best you can.

Ideally I would like to go ahead and get it under contract and sell it to an investor with a provision in the contract that the owner maintains possession of the property until a specific date.

Always better to have your financing options ahead of time .