I'm feeling a little shaky on this one. I want to take a stab at it, but someone who knows appositives better might come along and correct me, in which case I'll recant and go hide in a corner.
An appositive phrase is a noun phrase that renames another noun or noun phrase right next to it. Example:
Martha, the president of the company, called her assistant on her mobile phone.
"The president of the company" is the appositive.
Now let's look at your sentence:
She is thrilled to use her writing talents to support her favorite
causes, education being chief among them all.
Yes, this does appear to be an appositive phrase. "Education" renames "favorite causes." And "being chief among them" is a adjectival participle phrase modifying "education."
Your second question was about adding "with" to the sentence, like so:
She is thrilled to use her writing talents to support her favorite
causes, with education being chief among them all.
Yes, this is equally correct, though now it no longer qualifies as an appositive. It's just a run-of-the-mill adjectival prepositional phrase.