My wife and I want to buy another house in the same city using my second-tier entitlement.

My question is this:In negotiating with this lender for the second mortgage, how likely am I to be successful to work out a deal before the auction date?

I'd suggest option #1, but there's a second step that you haven't addressed.

She asked for a refund for that night, and I gave it to her, but I also refunded her the second night, even though we got the air conditioner fixed. 

Does it make sense to get a home equity loan on my first rental house to buy a second rental property?

Second, I am afraid the price of our home right now is at its peak estimated $635k and would like to use that equity money (roughly around $200k) to fuel my planned venture to financial freedom through real estate investing.

If you know who wants to sell and who wants to buy, you'll have the makings of a game plan in place that is second to none.

Here is a link to an interview I had with my father. 

My father has been involved in commercial real estate for about 40 years and has shared some of his knowledge with me. 

My parents got divorced when I was in the 5th grade but he kept me in his life and tried to be a regular father that made sure I did certain things like going to and graduating from college.