My question is - can we rent out my primary house after refinancing the primary house mortgage loan in less than a year without breaking any rules.

Just something else that could break and doesn't bring increased value.

Just to name a few there is OSU college which is a real estate super power that develops land and buildings at a crazy pace, L Brands (Victoria Secret and Bath and Body Works), and Battelle a well-known large government research facility. 

Most Hard money lenders are going to want 1st lien position and even if everything breaks right and a lender will take second position, a 40k loan amount is so small it eliminates a ton of options- An Asset based Lender at 75% to 80% LTV is your best option if you can not get conventional funding

-I suppose you could offer the tenant cash to break the lease early, still giving them some time to find another place... may not be a popular option but an option none the less.

I would simply make a 2 week, 1 month, 3 month, 6 month plan of a break down in what you want to accomplish and how.

I assume if you ask your lender for a no cost no fee refinance the rate will be 4.875 with APR 4.875  Ask for the APR which breaks down the cost of the rate averaged over first 12 months to get level numbers.

One solution is to let the lady break the lease and leave in exchange of my keeping the last month rent and security deposit.

It breaks even or I may even pay a little out (while also upgrading the house a little) throughout the year.

If you find the home that works for you, see what it will cost you to break the lease.