Re: Burning Lamp Oil
I'm an easy going DM who doesn't like to rule against player expectations as often as possible (I mean, who knows what Lamp Oil is made of in tge Forgotten Realms, anyway) but traditional real-world lamp oil wouldn't burn unless it was very hot. It wasn't kerosene. More like vegitable oil. Knowing me, if you try it, I will allow it to "work" but probably not as effectively as you would prefer. It'll light, but it won't be a raging inferno or anything.
Re: Eko. No one's seen him, but I don't think that would last if he tries to go through doors. The place is on High Alert and they are securing doors behind everyone. I'm going to say he chooses to fail getting through the door, rather than getting seen doing it, though.