Will I get my visa after I leave the country and try to come back with an FTA for my 2nd DUI?

I am not sure you would get VD with second DUI record.

Can a daughter who is a UK citizen residing in Hong Kong be an executor for a Florida will ?

Yours is not a question of US immigration law. I changed the practice area so you get better answers.

Pending security checks after n400 interview

It is not normal but you have two options: 1. wait and wait again. 2. seek federal court review.

I have a Green card through Asylum. Can I renew my old passport strictly to travel the world? Will this invalidate my status?

Can I renew my old passport strictly to travel the world? Will this invalidate my status? No, and I am afraid you already know that.

I 130 Petition for Alien relative has been filed with priority date of July 10 ,2015. When shall the review will start from USIS

Why USCIS? Was the petition not yet approved?

When it is best time to withdraw my pending asylum case? Before or after leaving USA?

Before, of course. If you are no longer seeking asylum, why to have the application pending.

I-131 RFE

It is true. Just answer the RFE. No need to be worried at this point.

Filed an I-539 for my mother, which is pending, and I-94 expires in December 2019 does she need to leave the country?

She can stay until I 539 is approved or 180 days pass from the point her I 94 expires whichever comes first.

This is not employment. This transaction is fine on H1B.

Has the participation in the green card lottery any negative impact for future visa applications?

It indicates your desire to immigrate to the USA.