Then observing and guiding tenants back to the terms of the rental agreement if they stray.

Showing: If the prospective tenants pass that stage, we invite them to see the property, at which time we observe and find out more about the prospective tenant.

Over that two years, I have observed they keep their aparments in bad condition, are very loud and bothersome.

And sites like padmapper and zilpy, and zilliow and Craigslist help to do that survey; I bookmark the search for those so that when the time to increase comes along, I click the bookmarks and observe the competition in the market. 

I've instructed the tenant to file a report if any illegal activity is observed and to take photos of the issue and send them to me.

Have you observed them walk; if so, how is their gait pattern? 

Also, for moral compass type questions an in-person interview would be best so both the verbal and non-verbal communication can be observed.  

Be prepared to observe:* Demeanor, interpersonal skills, communication style.

From what I have observed, often claims of big appreciation are related to LISTINGS and not SALES, and cashflow claims may be one-off outliers vs averages.3. 

I can see, clearly, you have seen some red flags and are leaning towards not renting them.Furthermore, I sense, per my observation of this thread, a bit of anxiousness in renting out the unit quickly.