What thickness of leather should I use?
For the SCA, if it's fighting armor, rather than decorataive, that depends on your Kingdom requirements, and what that armor is covering (bone or flesh). Different parts of your body will require different thicknesses (and don't forget the padding underneath). If it's not meant for actual combat, then it depends on how heavy you want it to be, and what you'll be doing with it. (Vague, I know, but we need more info!)
How do I dye the leather?
Do you mean all one color? Or multliple colors in different designs? You may be able to purchase complete hides in a particular color, which will save you a LOT of time, but you will most likely have to purchase bottles of dye. You should not use a water-based dye (i.e. Tandy's Eco-Flo) on your armor, because your armor will see a lot of moisture, and that water based dye will bleed and run.
What is the most effective way of hardening the leather that wont burn me or my house down?
Sorry, I have no experience with this. You might want to ask this question in SCA forums, instead of leatherwork forums.
Where is the best online supplier of cowhide and accessories?
That's hard to say--how much hide do you need, what weight, and so on? It might help if we knew what state or city you were in.
What tools should I use to get started?
Well, are you planning on doing tooling? Or just basic armor work?
I'll assume you mean the latter. So, for basic armor construction:
1) You'll be working with heavier leather, so you need good strong SHARP knives to cut the leather. Leather shears might work, depending on the thickness of your leather.
2)You'll need a metal ruler.
3) You'll need something to punch holes with. You'll be better off with a set of drive punches, rather than the plier type (which will only punch holes close to the edge).
4) Your drive punches will require a mallet--not a hammer, but a soft faced mallet.
5) A strap cutter would be very useful. You can live without one, but they make strap cutting FAST and easy.
6) Rivets for leather--there are different types, for different uses. Some are strong enough to make hinges, some are not. Make sure they are long enough for your leather thicknesses.
7) Rivet setting tools
8) Some kind of cutting board or surface to use, so you don't ruin your tools OR the kitchen table!
9) Some kind of surface for punching/pounding on, so you don't ruin etc.
10) If you are dying, some kind of spirit dye, with sponges or daubers to apply it.
11) Some kind of surface treatment to seal against dirt/moisture/etc.
That should do for starters.
If you are doing decorative stamping or tooling, you need to start with the above, and then we need to have a better idea about what you are planning to do before we can recommend tools.
I used an Xacto knife and a modeling tool for tooling for years before I decided to get serious.