Is the phrase "I love playing guitar, been doing it since a very young age" grammatically correct?

My quarrel with the phrase in the question title is the use of "been doing it". Is it valid to use it this way (without the word "having" in front of it)?

If it's not grammatically correct, is it at least informally acceptable?

This is perfectly grammatical, and perfectly common to boot. It is a simple ellipsis of "and have". It is not an ellipsis of "having", though. The former just lists two things. The latter introduces a relationship between them.

It is not the highest register, and is most likely to be encountered in speech, or in reproduction or rendering of speech in writing.

A quick search of the Corpus of Contemporary American English backs me up on both accounts.

• I shake my head, check the time. "I smash cars."
  "Pardon me?"
  I take the seat. "On Tuesday nights, I smash cars. Been doing it since high school."
  "Really?" Her eyes light up.

  — The Fourth Kiss by Michael Libling
  FANTASY & SCIENCE FICTION, 2002

• This is a guy who every Martin Luther King Day plays, uncut, the "I Have a Dream"speech. Been doing it for years. That's not the act of a — of a bigot.

  — CBS
  SPOKEN, 2007

• I'm a total rookie in television, only been doing it for a couple of years but I've been in radio for over 30 years.

  — Fox, BECK for April 15, 2009
  SPOKEN, 2009

• "The folks over at the fire station blow it every night at 6 to tell the townfolk it's time for supper. Been doing it for years," my aunt explained as I sighed with red-faced relief.

  — TEXAS MAGAZINE; Pg. 4, A very confusing trip to the coast
  MAGAZINE, 2002

Here's a direct link to my query for further examples. I am confident, though I haven't checked, that the British National Corpus and the Corpus of Historical American English will have further examples still, going farther back. Meaning to say, it is not a recent feature of the language, and is not limited to the New World.