Yes there's a tube station called Becontree, and there's a Becontree Heath as well, but most people would just say it's all part of Dagenham.
Depends on your point of reference I guess, if you were talking to someone from outside the area you'd probably just say Dagenham as it's better known. If you were talking to a local you might say Becontree to be more specific.
Adjacent. The District Line map will show you.
Both are part of the London Borough of Barking & Dagenham.
Becontree is partly in Barking and partly in Dagenham
No, but they are very close to each other