The header for a source file should define the interface that the users of the code need to use it accurately. It should contain all that they need to use the interface, but nothing extra. If they need the facility provided by xyz.cpp, then all that is required by the user is #include "xyz.h"
.
How 'xyz.h' provides that functionality is largely up to the implementer of 'xyz.h'. If it requires facilities that can only be specified by including a specific header, then 'xyz.h' should include that other header. If it can avoid including a specific header (by forward definition or any other clean means), it should do so.
In the example, my coding would probably depend on whether the 'foo.h' header was under the control of the same project as the 'blah.h' header. If so, then I probably would not make the explicit second include; if not, I might include it. However, the statements above should be forcing me to say "yes, include 'foo.h' just in case".
In my defense, I believe the C++ standard allows the inclusion of any one of the C++ headers to include others - as required by the implementation; this could be regarded as similar. The problem is that if you include just 'bar.h' and yet use features from 'blah.h', then when 'bar.h' is modified because its code no longer needs 'blah.h', then the user's code that used to compile (by accident) now fails.
However, if the user was accessing 'blah.h' facilities directly, then the user should have included 'blah.h' directly. The revised interface to the code in 'bar.h' does not need 'blah.h' any more, so any code that was using just the interface to 'bar.h' should be fine still. But if the code was using 'blah.h' too, then it should have been including it directly.
I suspect the Law of Demeter also should be considered - or could be viewed as influencing this. Basically, 'bar.h' should include the headers that are needed to make it work, whether directly or indirectly - and the consumers of 'bar.h' should not need to worry much about it.
To answer the last question: clearly, headers needed by the implementation but not needed by the interface should only be included in the implementation source code and absolutely not in the header. What the implementation uses is irrelevant to the user and compilation efficiency and information hiding both demand that the header only expose the minimum necessary information to the users of the header.