Who will configure the IP in Router A? Me or a ISP?
There is a very basic notion here. The person that owns a thing has responsibility for its configuration.
If you own Router A, then you configure it.
If your ISP owns Router A, then the ISP configures it.
I can't imagine a IT world which operates otherwise and actually works reliably.
"Stub Network" is a network accessed by a single router!
Incidentally, the network behind Router B is not commonly called a "stub network" in English. It's just called "my company's network" or "the network behind Router B".
When you say "stub", that carries a loaded meaning to network professionals, particularly since OSPF has stub networks; so does EIGRP