What does “walking middle finger” mean? Is it popular English word?

Solution 1:

In Western culture giving someone the middle finger - by making a closed fist and extending the middle finger into the air - is a curt way of communicating, Fuck you or Up yours or Screw off.

The phrase walking middle finger implies the person in question inherently is sending a "screw you" message to a group of people due to both her prominence and rhetoric.

As to your question on the use of hating on, it is not grammatically correct, as you note, but it is a phrase that has entered the slang lexicon. The Online Slang Dictionary notes the following meaning (among others, which I've omitted for brevity):

hate on

verb - transitive

to insult, complain, or criticize.

As an aside, Googling "hating on" returns roughly 3.5 million results, indicating it has entered the vernacular. Interestingly, Google Trends shows the phrase had no prevalence until early 2009. However, I'm unclear as to the origin of this slang.

Solution 2:

"Middle finger" is a reference to a rude gesture meaning "F- you", at least in American usage. It's usually used with a verb, not just as a noun phrase, and in that usage "middle" is understood: "she gave him the finger".

"Walking X" means being in a near-constant state of X. That person who always messes up projects he's assigned to might be a "walking disaster", the person who's always grumpy and angry might be a "walking storm cloud", etc.

So putting these together, the article is saying that Palin shows perpetual contempt for the BosNYWash elite.

"Hates on" strikes me as odd; I would just say "she hates the media".