Wilson - there's always a lot more to football than what's in the LOTG, just like there's a lot more to how the laws are interpreted and applied than what's written.
AR's do have all those duties - but the first line states 'subject to the decision of the referee'. So, by that, a ref can tell the AR he doesn't want him signalling a damn thing, and that's within the laws.
Most refereeing practice has evolved into what works best, through trial and error.
As for the 'jurisdiction' thing, any referee will know - probably better than anybody - just how important it is to be in the right position to make a call. What looks like a blatant foul against the red team from one angle may in fact be a blatant foul against the blue team - I've had times on the line when the ref has given a FK against the team who I thought got fouled, thus I can only assume he saw something better (and usually when I query later, this is the case). Of course it's frustrating to the spectators, but like them, the AR is basically stuck with a single view, whereas the ref can roam all around the field to get a better angle.
That's one thing I notice on the line - spectators will understandably fire up about a particular incident, but through experience of looking at fouls it becomes easy to see when a particular angle may be deceptive.
Then, naturally, when you're too far away from play it makes it harder to spot what's going on (though it's also possible to be too close to play - if players are a yard or two away, you may see nothing but a blur).
That's why, the further away the incident is from the AR, the less likely it is that he should call it. Even right on front of the AR, if the ref has a clear view and is fine with it, the AR should generally leave it for the ref (though the teamwork here varies between officials. Some refs will always wait for the AR's call in his 'area). If it's in the middle of the park, or the far side, then the AR has to acknowledge that his view probably isn't all that great, and the ref probably has a better one, thus should only flag if he thinks the ref has completely missed something (say, something off the ball), or the ref is looking over for help.
When I'm reffing, I always remind my AR's that the sort of things I'm likely to miss that are closer to me are pushing in the back, shirt pulling and deliberate handball - all things very easy to miss if you've got the wrong angle. For the AR to intervene when it's reasonable to assume the ref has a good view, when the AR knows his view isn't the best, and when the ref isn't looking for help...well, that's when he starts to interfere. The AR is only there to assist the referee, he isn't there to be a ref. The AR also has to call things the same way the ref is - if the ref is basically letting all pushing in the back go, then the AR has to apply the same standard.
So there are times when the AR is sure the ref's missed something, but also thinks the ref's had a perfectly clear view and a good angle, so generally he should leave it to the ref. But if he thinks something completely blatant has been missed......remember our GF against newcastle? It was the AR on the far end of the park who told the ref there was a deliberate handball. You'd only call something from that far away if you were absolutely certain (in those cases, 1% of doubt means you don't call it), but the AR did the right thing there.
Where to draw the line between assisting and interfering is a very fine one, and is one of the reasons why being a good AR is actually a really hard job.
The last paragraph about agreeing with the ref's signal, I don't agree with it, but that's what we're taught (we all go together, we look like a team, and it looks more convincing and certain than contradicting each other). If you have good officials though, then the ref will only signal first when he's certain he's seen a deflection the AR may have missed, and in that case the AR should assume that's what's happened, so generally the idea of 'going with the ref' works well, but there are times that it does allow the wrong decision to be made.
But then again, even if the AR was allowed to disagree with the ref, the ref still has to figure out why and decide who had the best view - so the wrong decision is still going to be made, probably just as often, except the players will complain more about it "look at your bloody linesman ref, he said it was our ball!".