I'm not sure I've received the e-mail messages you mention, unless it was a long time ago. And for some reason I don't think I got notification of this message of yours in the usual way. So I'm sorry I'm so late with this response.
Anyway, as far as I know, the surname Babianski is at least not rare in Lithuania. Whether it's as common as Smith in the U.S. I have no idea. The given name Konstanty is quite common in my own Babianski line. I don't know of any connection between the Babianski and Rimkus families, and while I don't have any reference to the Stankieck family, I suspect you've noticed from my line that I do have a Wincenty Babianski marrying a Klara or Anna Stankiewicz in or before 1816. While that seems too early for your records, it at least suggests a linkage between two families with names similar to the ones you've got.