Is it considered correct?
From a strict English language point-of-view, probably not. However, as it is more than likely done deliberately to achieve a specific effect in the reader's mind, author(s) of such constructs are probably not that bothered.
At heart, it's probably a comma splice error over phrases using Situational Ellipsis
Situational Ellipsis (Subject pronouns)
When we do not need to mention someone or something because it is obvious from the immediate situation, we use situational ellipsis.
We can also omit a third person pronoun (he, she, it, they) at the beginning of a clause in informal conversation when it is obvious who or what we are referring to.
Jessica felt the back of the note, rubbed the surface for coded dots.
with the subject pronoun restored would become:
Jessica felt the back of the note, she rubbed the surface for coded dots.
and fixing the comma-splice you might get:
Jessica felt the back of the note and she rubbed the surface for coded dots.
See also "Bill J" quoted in this answer about omitting a repeated subject.
Does this type of structure have a name?
If it does have a name (when done intentionally for specific effect) I don't know it, although I have seen similar constructs in several works of fiction when the author is trying to impart a sense of urgency or immediacy to the actions described.
To my mind, the "abbreviated" sentence form gives a sense of urgency -- almost as though the author hasn't time to describe the action "in full" without getting left behind. For example, taking the (hypothetical) three-clause phrase from dmzza's answer:
Jessica felt the back of the note, rubbed the surface for coded dots, deciphered what they meant.
This version (to me) imparts much more urgency of action than a more wordy, possibly more correct, version would do:
Jessica felt the back of the note. She rubbed the surface to check for coded dots and on finding some she deciphered what they meant.