The only way I got the answers to some of those questions was to just go out and do it. As you go you will figure out what you need and what you don't need. I guess you could go out and spend $1500 on supplies for the season but unless you know what you are going to be using then you might waste half of that or just leave it till the next year. I know that your question was meant to help you avoid wasting or buying something you don't need, but it just isn't something that someone else can tell you. You have to look at your market and what you plan on offering. Then stock your supplies to suit that.
As for advertising, I just use word of mouth. Its not my full time job so I am fine with getting jobs when I can. I have a couple people that are being patient enough to wait for me to get moved before I do anymore cars. I've got 3 or 4 lined up for the first week I'm done moving. Word of mouth is the best advertising you can do. It also helps you control your volume. I don't have the time to handle the volume I'd get if I advertised properly. If you aren't doing this as a hobby then I would say that you should get some business cards made up and some flyers and just leave them everywhere. I know several part time detailers that do that and it seems to work fine for them.