I would like to decrease my interest rate as well, which should save me money even in the short term.

So far, I have seen a net decrease (at least for the two in question).

Now when I just pull out my phone my muscle reflexes pull up yHE BP app.

On the other side, as demand decreases, home values have to be dropped by the owner setting/accepting a lower offer. 

. # beds/baths5. school districts6. historical neighborhood turnover ratios+ moreThe increase your monthly rents by decreasing your deposit, charging pet rent, tenant pays first $x of repairs + more....In today's market you cannot think like a 1980's property owner....

You decrease the incentive for somebody to sue you.

Would you savings significantly decrease after a down payment on a property? 

So in the example, wouldn't I be paying less and less interest every month/year , given the balance in continually decreasing? 

After you stabilize the property you will have forced the value up and can refi or heloc it out.Every dollar you decrease your expense's is a dollar that goes straight to you NOI.

Some of the lenders will roll the points into the loan which decreases your initial costs.