From the sound of this, it seems when Mr. Elliot set out to woo Mrs. Clay away from Sir Walter, he did a better job than he realized. Just for that, I hope Mrs. Clay manages to persuade him to marry her after all.

To answer your questions, if Mr. Shepherd is even semi-respectable (and we have no reason to think he isn't, despite his talent for flattery), he will disown his daughter as long as she continues her relationship with Mr. Elliot (though perhaps leaving the door open if she should ever leave Mr. Elliot, or on the other hand, marry him). The Shepherds will *not* be going to London to see their daughter's elegant new house (supposing she has one), visiting back and forth, or even mentioning her name if they can help it.

I think her father would automatically retain custody of the children (unenthusiastic though he might be at the prospect), because Mrs. Clay would be considered an unfit mother and a bad moral influence by the standards of her time (just as Kitty is carefully kept away from Lydia's influence at the end of P&P, even though Lydia is now legally married).