I cooked two chickens for dinner today.
I can say that, if I cooked this:
However, if I cooked different parts of chicken, I would treat the word chicken as an uncountable noun:
I cooked chicken for dinner today.
I would say that, if I had cooked something like this:
Interestingly enough, the uncountable expression works in either case. In other words, even if I cooked the two chickens in that top picture, I could still say:
I cooked chicken for dinner today.
The present perfect conceptualizes the past thing from the point of view of its relation to the present.
Having eaten pumpkin pie, I know what it tastes like.
My present knowledge is the result of eating pumpkin pie at some point prior to "now". The eating might have taken place 1 second ago or 1 year ago or 10 years ago. I might have some unfinished pie on a plate in front of me. All we know from the tense is that I am someone who ate some pumpkin pie at some time in the past.
With "having lived", if you said this and only this:
Having lived in India, I am familiar with the cuisine.
all that can be concluded from the statement is that you lived in India at some period in the past. It could be last week, or last year, or for the past 20 years. You are a person who can truthfully say "I lived in India". You might live there still.
Now, let's see if we complicate things by adding "all my life".
Having lived in India all my life, I am familiar with the cuisine.
All my life does not exclude the present moment; nor does it necessarily include it. You can easily say, as someone who has emigrated to Antarctica, "I lived in India all my life. But now I live here at the South Pole. Aren't the freezing temperatures refreshing?"
So, we cannot conclude from the statement whether you are living in India now, or have moved elsewhere. All we know is that you are a person who can claim to be someone who, at some time in the (recent) past, had been living in India since birth. all my life does not exclude the present but it does guarantee it: a person who had been born in India and who had lived there until very recently, could say "Having lived in India all my life..."
So, yes, you can say "having lived" whether you have moved away, or you still live there.
Best Answer
Your example 1 is a reduced subordinate clause. The full form would be "When she was asked about her parents, Ms. Angelina said she was uncertain of their names and promised to look them up..." Accordingly, "asked" is part of a passive construction and is the past participle of "ask". (Personally, I'd prefer to call it a "passive participle".)
Similar to the "whiz" reduction of relative clauses, "she" referring to the main clause subject and the form of "be" can be deleted, giving "When asked about her parents, Ms. Angelina ..." Additionally, "when" can be left understood (as other subordinate conjunctions can be, sometimes).
The other two examples are only marginally acceptable, in my opinion.