Welcome to The Holy Land of Hoops

Like the title states, if we are getting the Big East name, do we get the conference history as well?

Short answer... Yes.

I'd say it sticks with the name. Also, logistically, it makes no sense to give it to the three remnants, two of which were rather recent additions themselves.

so does the new BEC also get:
the website
headquarters
sponsors

This Big East is a brand, everything associated with it will be included in the deal, unless otherwise negotiated. The C7 aren't going to be paying millions for just two words.