Honestly, I've never heard the superstition that you said about old people, but it sounds pretty interesting :)

It's called "a caul" in English. I, in fact, was born with a caul myself, according to my mother. Apparently, there was once a superstition that it was good luck to be born with a caul. Furthermore, there was a superstition among sailors that anyone who possessed the caul from a baby could not drown. in the opening passage of a famous novel by Charles Dickens, "David Copperfield," the narrator says: "I was born with a caul, which was advertised for sale, in the newspapers, at the low price of fifteen guineas," hoping that a sailor would buy it! The correct name for what you call "the bubble" is "the amnion" or "the amniotic sac." When it breaks during childbirth, in English it is said that "her waters broke." (For some reason, "waters" is plural.)

*...the baby is inside the mother's womb *Sometimes during the process of giving birth... *...part of the womb may stay on the baby... What a strange and interesting expression! My Russian students often tell me about common superstitions that continue to exist even today. There was one about not telling an old person the time when they ask you for it...apparently it's because they are trying to steal your youth! We don't have this expression in English, at least not that I'm aware of. We sometimes say someone was 'born with a silver spoon in their mouth' if they have rich parents who buy them everything they want. We also have many superstitions of our own, though I don't know of any related to childbirth. At least, none that are really interesting.

Ive never heard of any such thing in English.