Absolutely not. OT's inboxes (and other boxes) are integral to the internal workings of OT. (And everything in them is signed.)
Bitmessage is an additional layer to be used for discovery, specifically for server-to-server wiring and cross-server exchange. It will be used with OT, inside this GUI (this thread), which will integrate with both of them.
Absolutely not. Bitmessage will not change anything inside OT itself. Rather, this "holy grail" application will use both OT and Bitmessage.
Also, FYI, the Bitmessage communications will not be between OT servers, but P2P between OT users.