No one knows your stress level you can handle or want to handle but you.

As an agent, I handled all the negotiations and paperwork regarding the offer and the post closing occupancy.

Would I need to have a loan servicing company handle the payments for me if there was no mortgage on the property?

Most of the landlords that I know around me that do something similar to what you're talking about find a good handyman that they can just send their issues to when they arise and then just handle everything else.Just some of my thoughts!

In most circumstances for single family homes, the investor/owner will require the tenant to handle landscaping and snow removal. 

But we are taking possession of the house this week and we will not receive rent this month since it has already been paid to the current seller.How is this situation typically handled?

Is there a better way to handle it? 

If your property manager is handling it in a a timely fashion, let them do their job. 

We self-manage our LTRs but are open to letting a company handle the STR.

They sound small, so it's likely that there aren't brokers focused only on them, but rather will be covered by brokers in nearby larger cities, especially since you are talking about commercial properties like hotels and multifamily, and these tend not to be handled by your local residential brokers.