I talked to Emerson and he said to go on in. I'm standing at the entrance and no door pops up. What am I not doing?

There should be three guards blocking the entrance to the mines. After you talk to Emerson you should talk to the middle guard of the three. Two of them will leave, but the middle one will always stay there. Just walk around the remaining guard into the entrance.

I'm at the cave entrence and there are no guards at all. I am standing at the place where the tracks go into the side of the hill. Am I doing something wrong?

That's the wrong door.
Go over to where Emerson is standing and look a little up. You should see three guards standing at a door. Talk to one and ´he will wish you good luck and let you go into the mine.

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"No!" shreiked Tyrion. Though the sword felt heavy as a fallen tree he lifted it. He saw the deamon look up, astonished. He brought down the blade with the awesome majestic power of a falling thunderbolt. The burning blade caught the deamon right in the centre of its forehead. The deamon's head fell apart under the force of the impact. The deamon N'kari sank to its knees, dead.

-Tyrion at the battle of the Blighted Isle.

I have heard that if your rep is too low and/or the character doing the talking has too low charisma they won't let you in.

My reputation and/or my charisma is to low. How dare you...you...you cad you! It sounds like your turning me down for a date. No; but seriously, I was at the wrong place. It was kind of anti-climactic. Little kobolds aren't that tough; but, the atmosphere was pretty good though. I kept expecting something more deadly with all that music. At first, I went out the cave in, and boy, did I get smoked. I didn't go out that way again. I'm gaunna' get those suckers! Thanks for all the help. I'm really having fun.