There’s a book by Gavin de Becker “The gift of Fear” that has some good advice on dealing with people who make threats of any type.  

That's a huge gift brother, so don't take it for granted.So about your credit score or lack of.

Is there a way to use that “gift of equity” as down payment, or any other options to help us out and save some money.

Listen to the seller and clue in on the reason they are selling.if she won't call back or answer your calls, send a nice letter apologizing (if you think you have some to apologize for) or just something saying "thank you for your time, i would still like to talk to about the sale of your home"....maybe a thank you card for her time and consideration......a gift card or offer to sit and buy her a cup of coffee...... again trying to resolve any issues that may lead to you closing this deal....you worked hard for this lead (i presume) dont just give up!....

When he does this, he may choose to report gifting you the half.

But also, my wife and I floated the idea to let our children slowly buy their way in as they grow into teens and young adults.The IRS says I can only gift my children so much money, and I don't really want to gift them anything.

The IRS does not care whether it is a conventional loan, private loan, gift from a friend, sale of stock, savings, robbing your child's piggy bank or wherever you have access to money. 

I’ve written her an extensive text message and had a visa gift card i Usually give tenants.

He declined unless we wanted a steep discount and then gifted them so funds back to save them taxes.  

For instance, I had a family friend that gifted their home then had a sudden illness that required long term nursing home care.