Learn English – “to have something being done” or “to be having something done”

The first two are grammatical and come across as idiomatic. The third does not.

In the case of Being 19 weeks outside the womb my baby has learned to grab things, you are stating that your baby is currently 19 weeks outside the womb and that, as a consequence, it has learned to grab things. Being 19 weeks outside the womb is a state that your baby currently is in.

In the case of Having been 19 weeks outside the womb my baby has learned to grab things, you are stating that your baby, at some time in the past, was 19 weeks outside the womb, and, as a consequence, your baby now has learned to grab things.

I hope that clears things up!

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.