1) Is it the actual "closing date" of the new property that needs to be done in 180 days?

PS I had to put up the entire purchase price in CASH because that's what my contract said I was doing paying cash  into escrow on the close date and let it sit there their entire time.. and it was over 150k..

Buy a new stove and increase her rent to pay for the stove, you purchased. when her year date comes up just tell her the rent increase is due to maintaining the properties conditioning and amenities

When’s the Feb date?

It means 30 days, which could start or end on any date of the month.

It can just be a one pager stating the important facts "purchase price, deposit, close date, conditions, parties etc.." then where you get serious have a lawyer draft out an agreement.. 

One of the terms states "Quick Closing * – Buyer must be willing and able to execute assignment contract by set date" Not sure what this term is stating with "execute assigned contract" .

The facts and challenges are-1) The house is listed for sale on the MLS and it’s “short sale listing approved by bank” leaves no deal. 2) The first lien holder, which is a FHA loan, is for foreclosing and the sale date is set for about 3 months from now. 3) The owner died and his estranged daughter now has the right to sell. 4) The estranged daughter has zero motivation to engage in the process. 5) The lender is owed about what the houses ARV might be after $75k in renovations and carrying costs. 6) I’ve been able to connect with the attorney handling the foreclosure in the interest of pursing buying the note who reports that since it is FHA they are unable to accept a short payoff.

You should have one on speed dial. 

If they accept, coordinate with them a date and time along with a locksmith so as soon as they are out of the house the locksmith can simultaneously change all the locks to the house.