You can grapple multiple targets, but only until your turn.

Looking at the rules, I see nothing that disagrees with your idea that the witch can start extra grapples with her hair using attacks of opportunity while she grapples someone else. As you note, it takes a standard action to keep someone grappled, which means that unless you find a way of generating extra standard actions, you can only keep one opponent grappled once it's your turn. The others go free.

Your DM is correct: it does provoke an AoO.

The confusion here comes from your reading of the sentence where you believe that the second part only applies if the first is true. The sentence is in fact a two item list; the first half and second half do not interact with each other in any way.

The d20pfsrd also says

A spell-like ability has a casting time of 1 standard action unless noted otherwise in the ability or spell description. In all other ways, a spell-like ability functions just like a spell.

And spells provoke (except when cast as a swift action).

Interesting, you can Arcane Bloodrage defensively, since you have not actually entered a bloodrage when you apply the effect, it is done as part of the activation.

This is the RAW on the matter. The fact that swift action spells do not provoke does not mean that free action spells or spell-likes do not, though it is certainly worth making that point when discussing this with your DM. (Especially since the reason is that there are no free action spells any more is, to the best of my knowledge, because when pathfinder came out it had turned them all into swift actions)