This gives you a legitimate excuse to tour every room in the property during the lease without being too intrusive.

What are your thoughts/recommendations for...1) Key-less entry2) Ring door bell (too intrusive?)

Could it be water intrusion from the outside at all?

Depends on how bad the water intrusion really is.

Water intrusion is a PITA.

From environmental standpoint, u may want to collect groundwater samples along boundary of the property with the gas statiob to see if it has impacted groundwater as well as soil gas samples to see if there is potential for vapor intrusion.if groundwater not used, then principal environmental risk would be vapor intrusion which can be effectively managed with radon system and vapor barrier.Note that if the gas station is downgradient from your property (gwvflows towards gas station), vapor intrusion can still be a problem since vapors can migrate thru sewers, utilities and any conduits.

Seems very intrusive.

That allows for water intrusion which leads to rot.

Are the questions too intrusive? 

Some things they have found that I wouldn't have known about is improper grading in the front yard which is causing water intrusion issues and foundation shifts and the electrical being out of code due to it's age.