I am planning to mail him a certified letter of requesting to clean the yard.

I'll send you a connection request and maybe can help answer any questions you might have for the fort worth area 

Sure, you can refuse the tenant's request to add cameras and change the doorbell, especially if it's laid out in your lease agreement. 

I've requested a ledger from the owner to verify the tenant's payment history, but I'm aware that documents and screenshots can be falsified.

What has worked for me very well is the fact that I almost never speak with tenants about repairs, at signing we go over the instructions in my rental agreement on how to go about maintenance requests, expectations and possible delays. 

Is it completely unreasonable to request another background check through a more reliable engine like Mysmartmoves.com? 

What you can do is request rental history & applications so you can review it yourself to find out as much information as possible in terms of their payment history. 

They should have simply returned your request denied and not shared any additional details or motivations.

This is my very first time renting the unit as I purchased it a couple of months ago.Though they are requesting application, but being out-of-state is not instilling confidence in me. 

I have seen traveling nurses with a partner and a child on some requests, or generally 2 parents and 2 children.