Privacy: I think a url with a long enough random identifier would be sufficient and is often used in web apps to share content.

Multiple TLDs: Since the tlds are quite short, I would keep the name on the left and include one column per tld (with the availability icon). If the user enters the name with the tld then only that column shows availability and the others have a "click here to check availability" symbol.

Maybe the app could learn what I like searching for automatically? E.g. you enter foo.com then foo.de, foo.ch, foo.at etc. Each time there is only one line 'foo' and four column (com, de, ch, at). The next time the user only enters a name it automatically checks them all.

Bug: You're right that is what I'm talking about and I didn't recognize I was filtering the list. Maybe simply adding "List filtered by 'xyz'" would clear that up.