[susan53]

Just = a very short time ago
Already =Before now but at some undefined time.

So - in all these examples "just" would be possible if the speaker had done the action a couple of minutes earlier. In (1) it's unlikely but not impossible. In (2) and (3) it's perfectly possible. It would depend on when the speaker had seen the notice/spoken to John. If yesterday, then "already"; if a few minutes earlier then "just" is equally possible. So basically they're bad test items. To clarify the difference you'd need to specify the times - eg I know, I've already seen it. David put it up yesterday and showed it to me. vs I know, I've just seen it. I was in the office two minutes ago.

"watched" is possible in (2) but less likely. "see" conveys the idea of a completed action - which is appropriate here, while "watch" conveys the sense of an on-going action. Compare :

1. I've already seen that film three times. I don't want to see it again.
with :
2. Jean says there's a point in Lord of the Rings" where Gandalf says "Rubbish!" but I've seen/watched it three times and I still haven't noticed it.

In (1), the emphasis is on the completed action - so see is more likely. In (2) the speaker might want to convey an idea of the total length of time spent viewing the film - so might choose watch.