How long should I wait for players to post before PMing if they haven't said they would be busy?
Generally, 3 days as this is plenty of time for most to post. If they have been active, them not posting for 3 days means that for whatever reason they may have missed the fact that the thread updated or that they haven't posted their action. If they haven't been active, PMing is still a good idea so they get to making the action ASAP when they log back on.
How long should I wait before automating the players that don't post?
There's no good time range, really. You could wait a few hours, a day, two days... But make sure that it's not something excessively long, like a whole week. Or excessively short, like a few hours. Also, be sure to PM them first.
A player has been inactive for a while (a few weeks), and has been dragging down on the game. Should I kick them out?
Yes. It's bad for everyone to wait for someone who may not post to post.
What rolls should I use?
Your choice. Generally, 1d6--or, in other words, a six-sided die--is the most common and easiest to work with due to simplicity. It may or may not be very merciful, or it may even be too easy, but generally it works. If it doesn't, or if it doesn't fit your fancy, you should try something else out.
Should I quote players' actions every turn?
It depends. It's messier and uglier to quote the actions, but it makes it
much easier and much less annoying for someone to see what someone else was trying to do.
What tools are useful to me?
www.random.org for various tools including a very basic dice roller.
www.wizards.com/dnd/dice/dice.htm for very good dice rolling tools.
http://lazarus.interclue.com/ for recovery tool. Trust us, you WILL eventually lose a post and will be very upset if you don't have it. A text editing program also works, but isn't even nearly as easy as this tool. All sites are safe, so do not worry about viruses.
Is a story important?
YES, a very, very basic story is a MUST if anyone is to know what the hell they are doing or what they should do. Complex stories are not as important, but at the very least a "X is doing Y, you are Z, some Ws, and you must stop X from doing Y" or your RTD will have, at best, a rocky start. This is all excepting RTDs that are just quick deathmatches of some sort.