Hmm, I was a little confused and have edited this post several times, but I think I have the hang of my answer now.
I think the first thing to note is that both the rules for Exhausted and Berserk have been errataed since publication, because they are both tricky. But I think how it works is now this.
(For ease of reference: Spending a favor and taking damage = paying with blood.)
About the free move: You can take a free move whenever you are activating, as long as you aren't exhausted. An exhausted model cannot take actions except for one exception: If you use Berserk to pay for it in blood.
So: If you are fatigued, and activate with Berserk, then you get to do a free move as your first action. If you choose to pay with blood for an attack instead, then you remain fatigued, so you get to do a free move after that. But if you spend fatigue for any reason, then you become exhausted, and once you are exhausted, the free move is forbidden. All you can do when exhausted is pay for a move or attack with blood.
So if you activate an exhausted model, the same principle applies. You get no free move, all you get to do is take one action, which you have to pay for with blood.
Other than that...
"If you activate a Berserk model that’s ready, then no effect."
Well, the second part of Berserk still works. So you can activate a ready model, and Berserk will allow it to attack three times, for example, paying for the last attack with blood.
"What I can’t do is power attack [with a fatigued model] as that would take my fatigue level beyond fatigued and the rules state you can’t go beyond fatigued."
I'm not sure whether you would be allowed to power attack in that scenario. Technically, probably no, since Berserk says you can pay in blood to "take an action", and upgrading an attack to a power attack doesn't seem to me to be an action in itself. But I don't see any problem in saying you can. The intent behind Berserk seems to be to allow you to spend three fatigue where normal people can only spend two. So being down one fatigue and still doing a power attack, meaning you are down three in all, seems to me to fall within what Berserk is meant to do.