As long as the Tenant is on a month to month, and proper notice is given, a rental rate increase would be an acceptable ask.Just verify that the appropriate notice period still falls within your escrow. 

I'm in favor of a short term lease (they can pick the period, but then you set the price) or another 12-month term with the standard rental rate.  

We've speculated that the offers may have been rejected because of the due diligence period being 7+ days, or perhaps being contingent on inspection, but have received no feedback from the agents as to the true cause. 

That is the only time I have ever heard of a required waiting period.

--An agreement to schedule walkthroughs periodically.

I figure this sort of strategy would only work for homes on the market for long periods. 

I would love to hear some updates about this property in the next few months.

I did major renovations on this house such as: new roof, new plumbing all over, new heat, all new windows, flooring, updated electric work.Purchased last March for 148k.

I purchased a flood policy today (there is a 30 day waiting period after applying for a policy) so my insurance will take effect April 1.  

You would typically want to have your contractor walk the property with you before you make your offer so you can factor the rehab estimate into your analysis and then you would have an inspection done immediately after you get the property under contract during your contingency period to make sure there's no other issues.