I'd definitely require criminal, credit, and eviction checks in addition to an application. 

I had tax returns, numerous free and clear properties, great cash flow, etc. but still had to personally guarantee - so yes, your personal credit and financial statement will matter.

I had an S Corp for my IT business, an LLC for my Auto Repair business, and a C Corp set up for my real estate management.While in my mind I need a real estate management business, when I did the papers to establish the management company, one question on the application is what type of business is the entity engaged in, and it clearly states if the company is engaged in anything requiring a license, I must submit a copy of the license, and I do not have a broker's license that entitled me to engage in property management.

@Kallie Krammes, lots of good nuggets above.You may also ask to review their rental applications. 

Here’s the applicable law from your state:TENANT’S RECOURSE FOR WITHHELD SECURITY DEPOSITSIf the landlord does not either return the entire security deposit or send an itemized list of deductions, along with any remaining portion of the security deposit, within the required time period, the landlord forfeits all rights to withhold any of the security deposit.

Most modern hotels are like this.Yes, I have had numerous issues with guests going to extremes with the climate control, and my property is located in an area where electric is very expensive.

We get many hits- 5+ inquiries per day and have done about 4-5 showings/week for the past month but NO applicants.

does anyone else experience potential tenants that don't want to fill out an application?

There is little downside as the no eviction rules only applies to applicants: https://www.houstonrentassistance.org/landlords

Do I use the same criteria as above and invite the first ones who applied & fit the bill to do the background and application check?