I was watching LAX, and somehow, a revelation came to me on a few mysteries. How LAX and these mysteries are related in my mind is, in itself, a mystery.
Purge - Well, we know that in 1988, Widmore was the leader of the Others. He was most likely until 1992 after the Purge. We also know that Widmore's goal is to exploit the island and it's properties. The DHARMA initiative was doing the same thing. Widmore most have known this. My theory is Widmore knew the only way to gain access to the research that DHARMA had done was to get isndie their camp and see their papers, research results, etc. So, my guess is he lied and told the Others Jacob had old him that they needed to kill DHAMRA. He most likely lied to the Others so they would think it was Jacob's order, when really it was Widmore's because he wanted to gain access to DHARMA'S work so he would be one step closer to exploiting the island for his personal gain.
Exile - If my previous theory is true, my guess is that somehow, the Others found out that Widmore lied and he wanted to exploit the island only. My guess is Jacob went to Richard after he found out of the Purge and told Richard of Widmore's lie. So they knew of Widmore's true intentions. You cannot be an Other if you only want to exploit the island, not protect it. So they diecded to exile him for his lie and his goal. How Ben relates to this and why Widmore is angry at him, well I haven't figured that out yet.
Supply Drop - The drops should have stopped after the Purge, correct. Ann Arbor should have known that DHAMRA was gone so they should have stopped sending resupply drops. Wrong. The Others must have kept contact with Ann Arbor after they wiped out DHARMA. They need supplies too and they know that if they pretended to be DHARMA folk, they could still get resupplies from headquarters. Ben writes in his diary "We are short on provisions (seasonal) and, more importantly, the long-awaited re-supply of camp gear having been missed (and so narrowly!)" This must mean they were expecting the drop, yet it was taken by our survivors.
These seem like reasonable answers to these questions except for that fact that no plane was visible when the supply drop came, which I'm still thinking about why that is.