If any risk becomes less of an unforeseen event and more of an inevitable one, the insurance industry will no longer cover it (think asbestos, black mold, etc...)

R&M and Capex would cover unforeseen items (redoing a unit, a boiler breaking etc.) as all the other expenses that are ongoing and paid for by the landlord on an ongoing basis would be in the financials provided and that I am going to make sure are repped to in the contract.

This deal also had plenty of unforeseen circumstances which also can be tabled for another discussion.

If you've done your homework and looked at comps in your market, have the capital for the down payment plus some cushion for vacancy, unforeseen repair and other costs then it's really just a matter of pulling the trigger.  

Yes, I sure did. it was to pay for additional administrative and/or additional activities required which were unforeseen at the time the lease was made, the agreed rents were also a tad under market rents and fees were not cumulative adding one infraction on top of another, so even with additional charges rents were not excessive.The basis was for the time required to address these unforeseen matters, charge was by the hour and the time spent or claimed was actual or my estimate and never excessive and was rarely used.The lease was approved under HUD regulations by/and our City Attorney for low/moderate income tenants under an economic development program where a "funny money" loan was assumed on the property, I bought the property from the city.  

It could all go toward unforeseen items and/or building a cushion?  

I'd like to save approximately 10% to cover unforeseen repairs, etc.

It won't even be accusatory, simply a story about how you solved an unforeseen problem.

The deal was we were to refi together but because of unforeseen circumstances I was unable to secure a loan relative took it upon them selves to start dealing with service provider thus avoiding me hoping to take over ownership I advised them numerous times that the service provider does not own the property I do. 1.

I don't think your idea to increase pricing for the new tenant and her dog is a bad idea; just make sure it takes into account that it is currently underpriced and that potential unforeseen damages can occur with pets.