It would be easier to read the sentence pair if 'here' was unambiguously differentiated from 'New York'—for example by saying 'in Chicago' instead of 'here'.
Having said that, all three (A, B, and C) can be correct.
A Jack lived in New York for ten years. Now he lives here (in Chicago).
'Lived' seems and is obviously correct, as we are talking about a completed past action that took place entirely in the past (even if it lasted ten years) and cannot include present time. Here the ten years and now are mutually exclusive times.
B Jack has lived in New York for ten years. Now he lives here (in Chicago).
Jack has at some point in his life lived in New York for a total of ten years in the past, either all at once or over several periods. And now he lives in Chicago. The ten years is all in the past and excludes now. These two statements are just two statements, and aren't necessarily connected other than by topic matter. Jack could also have lived in Sao Paolo immediately before moving to Chicago.
C Jack has been living in New York for ten years. Now he lives here (in Chicago).
The present perfect continuous or progressive can refer to recently completed actions that one is no longer doing. Example: I've been out running for an hour; now I'm home. Thus, the present perfect continuous can refer to the most recent ten years prior to moving to Chicago.
The real discord in your second sentence
I've never known my grandmother […]
is not so much the fact that you no longer have an opportunity to know her, but the fact that you used never.
A more concordant phrasing would be
I never knew my grandmother, since she died before I was born.
Even if your grandmother were still alive (and, for example, you simply never met her), it would be strange to use have known with never. The reason is that have known is typically used to predicate of unbounded (not countable) groups. A context in which we use have known with never is in phrases like
I've never known anyone to turn down an offer of ice cream after
dinner.
Best Answer
In many contexts (like this) both are fine.
Both of these indicate an ongoing situation that has now resolved. The present perfect indicates the situation just resolved. The past continuous does not provide any time frame, but it will often be obvious from context.
There may be a slight difference in nuance:
"I have been studying" suggests I may be studying now, or will keep studying after we finish our conversation. "I was studying" suggests that I stopped for some reason, and may or may not continue.