"Was Sie betrifft, bin ich zuständig."
I'm not getting this one. Does it mean something like "I'm for responsible for that which concerns you?"
The grammar in the English translation is beyond atrocious.
I suppose one way of looking at this sentence in English would be: "Concerning you, I am responsible."
Why is it in some senteces that zuständig is used, while in others verantwortlich is used? Is there any difference?
Yes, I would say there is a difference in German, but both "zuständig" and "verantwortlich" can be translated as "responsible".
zuständig reminds me more of a situation in some kind of office, etc. meaning something like "it's my job to do that" / "it's my job to take care of your problems".
responsible seems a little stronger to me, like "I take responsibility for that. If it goes wrong, you have to blame me" / "I am responsible for you, if you do something wrong, they will blame me as well".
Maybe that helps a bit.
Another translation that makes very little sense in English. such a sentence and with its aggressive sentiment simply would not be used. The nearest I got and I lost a heart, was 'What concerns you, am I responsible? However the translation seem more of an instruction or statement of a fact, about the work the person does and who is responsible for that work, rather than an inquiry about someones welfare.