Looking at her Twitter its very right wing indeed - so clearly she must have changed her views since her days in the Greens, given that I can't see how any one with that set of beliefs would have thought the Greens a natural home politically at all.
Her tweets appear to be those of a very angry person, hard to imagine she was every a Green councillor.
I struggle to see why that twitter joke is a sackable offence. I note that the report says "His personal Facebook account was also linked to various other EDL posts", and we don't know what was in those posts, but IMO that sort of post should have been dealt with by an apology and some contrition. Having said that, I could be convinced that sharing anything from the EDL could itself make the situation much more serious.
Edit: It also isnt made clear whether he was offered the chance to apologise and refused.