You are basically taking out a loan for personal use, just as you would spend money out of the credit card to buy a food. 

Fortunately, I have excellent credit, but I’ve been an independent contractor for the past 6 years, so I worry about qualifying for a mortgage.

You can’t even offer rent credits or require the tenant buyer to do maintenance on their property anymore

He's telling you to wait a week because you're going to go look at cars in person, or put your contact info on some website, to "think about it" a week earlier than whatever he tells you (this is 100% true, and as a lender I will not believe you if you tell me otherwise... sorry :P ), and the second you walk onto that car lot a bunch of car salesmen are going to descend upon you, deploying various lines of BS as a pretext to run your credit ("it's a soft pull" or "we just need your SSN to verify you aren't a terrorist, we aren't going to run your credit") and close you on a car that very day, and there is greater than a 0% chance you will believe what they tell you and screw everything up.The rule is that whatever you tell homebuyers not to do, some non-trivial percentage of them are going to push it one notch farther than you tell them.

Then you can pursue in small claims court or put a deficiency on their credit report 

If you have a tax company and real estate that is paid off, you should have no problems getting an unsecured line of credit.

Also, is it the first person who submits an application and passes the background/credit check who I move forward with as the tenant to fill the unit?

Your final considerations are Credit Cards or a HELOC but do their benefits carefully and pay them off in time. 

Examples: A guy with hotels who used business /personal credit cards for free flights, a guy doing private investing/hard money lending, and last month was a guy who told the story of a 12+ property deal spanning 2 states and 1 primary residence of the seller.  

(Avoid big banks, try local banks, credit unions, or mortgage brokers.)