Go with your gut. you wouldn't have come to BP for answers if you weren't feeling apprehensive about this.
Everything you posted sounds like they are flakes. What normal people with jobs live with their parents, leave empty spaces on important applications, and lie about their income/pay stubs? This is only the beginning of the lies.
We had a "nice" couple look at one of our sfh's and we had high hopes for them, tattoos and all. Polite, married, a couple of kids, and they filled out the app. completely. A simple search of the county clerk website came up with the following convictions between them: rape, burglary, aggravated robbery, assault, and fraud. You cannot judge the proverbial book by its cover.
The woman had the nerve to ask me how long she and hubby were going to have to pay for their crimes--the fraud had occurred a year ago, the rest within the last 2-7 years. Um, you kinda shoulda thought of that when you sociopaths were robbing, assaulting, raping and committing fraud.
So thankful for the brethren here on BP who opened our eyes really wide before we started this adventure.
Becky