On re-reading the BGC rules (pg 32, SG) I do see Qi foci are in kind of a unique place in how the rules interact. Marcus is right in that you'll want to discuss with your GM, I guess. Personally, *I'd* say a "reduced" Qi focus still grants all the power points (and you naturally suffer the BGC penalty to using those abilities) but I can see how another GM might read it as taking away power points (and ghost only knows what the priority would be on which powers are lost first...)
I'm hesitant to go offering more opinions without re-reading the rules, but as for centering my gut (without referencing the rules) is that you don't get around a BGC... if it's magic you suffer the penalty. Hard stop. (and yes, for each and every test, whether it's magic giving more dice [increased AGI, combat sense, etc] or some benefit to the outcome [Killing Hands, etc])
Edit: Adept Centering vs BGC:
Adept Centering
allows you to reduce negative dice pool modifiers to
Physical and Combat skills (such as modifiers from adverse
conditions) by performing some mundane action
at the same time.
A background count impose a negative dice pool penalty
equal to its rating for all tests linked in any way to magic (such
as spellcasting, summoning, and skill tests that use active adept
powers such as Killing Hands or Improved Sense). The exception
to this rule are background counts from domains, where a tradition,
person, skill group, or skill may be exempt from the penalty as they
are used to or aligned with the domain.
You have on one hand Adept Centering that says it cancels penalties "such as", and on the other you have BGC that imposes a penalty on ALL magic. Just weighing those against each other, I'd lean towards BGC outweighing Adept Centering based on semantics alone. But then there's the 2nd sentence, which gives explicitly gives exactly one exception to BGC penalties, and it's not Adept Centering.
So it looks pretty well beyond the realm of debate (to me, at any rate) that Adept Centering won't cancel BGC penalties.