The specs and pics tell a lot about the different gear posted and how it might work together. Generally speaking, most clamps / extensions / mic bars have either a 1/4"-20 thread, a 3/8" thread, a 5/8"-27 thread, or a 5/8"-male stud. Fortunately, it's easy to find adapters to swap between all those options. But, of course, connecting two pieces natively without adapters is often preferable.
Per the specs, the Bogen 237HD FlexArm has a 3/8"-female threaded base, so it'll fit any clamp that can accommodate a 3/8"-male threaded tip, whether natively or with adapters to convert 5/8"-27-male > 3/8"-male, or 1/4"-20-male > 3/8"-male, or 5/8"-male stud > 3/8"-male thread (like the
Bogen Rapidapter).
The BG649 C Clamp specs are a little more ambiguous, but from the pics looks like it has a 3/8" threaded tip on top of a 5/8" male stud. So it should either work with the Bogen 237HD FlexArm natively (3/8"-male on the clamp + 3/8"-female on the FlexArm), or would require a conversion from 5/8"-male stud > 3/8"-male thread (like the
Bogen Rapidapter). If, by chance, the threaded tip is actually 1/4"-20-male (doubtful, based on the following parenthetical comment), it would require an adapter to go from 1/4"-20 > 3/8"-male thread. (FWIW, the gold-colored adapter at the bottom of the pic looks to me like a 1/4"-20 thread on top of either a 3/8"-female or 5/8"-20-female thread. Can't tell since we can't see the bottom of the thread adapter, but I assume the adapter has a 3/8"-female thread for easy conversion of the native 3/8"-male tip to a 1/4"-20-male tip.)
As for the Adorama-branded SuperClamp, it has a reversible stud: a 5/8"-male stud with a 1/4"-20-male tip, or a 5/8"-male stud with a 3/8" tip. The Bogen 237HD FlexArm would fit on either - the latter natively (3/8"-male thread on the clamp + 3/8"-female thread on the FlexArm), the former with a 1/4"-20-male > 3/8"-male adapter, or either with a 5/8"-male stud > 3/8"-male adapter (like the
Bogen Rapidapter).
As for the Bogen MiniClamp, I'm guessing it uses the same reversible double-ended stud as the above Adorama-branded SuperClamp, so the same options would apply.
Finally, yes - the Mini Clamp Extension allows one to make a 90º turn when the extension's slotted into a SuperClamp. The tip of the extension is 5/8"-male stud, and what looks like a 3/8"-male thread. So the aforementioned Rapidapter would work for sure. But if it is, in fact, a 3/8"-male threaded tip, then no adapter would be necessary to connect the Bogen 237HD FlexArm. On the off-hand the extension's threaded tip os 1/4"-20, again, you'd simply need to convert 1/4"-20 to 3/8" to attach the FlexArm.
So, basically, you can connect the FlexArm to any of the clamps listed, it's just a question of how. The above's probably a bit confusing, and I mix up the different threads/adapters all the time, so feel free to ask clarifying questions.