I am doing a cake for a decorating competition the Tuesday after next and I was wondering if it would be fine to start baking and frosting the cake this upcoming Tuesday and then cover it in fondant? If it is should I leave it out or put it in the fridge?

I wouldn't bake and frost a cake that far in advance if people are going to be eating it. Too big a risk for it to go bad and make people sick or even grow mold. (IMHO) The most in advance that I've baked and decorated a cake is two-three days. If it's not to be consumed then that's another thing.

Why don't you bake, tort and crumb coat it and then freeze it. Just make sure you wrap it well. Bring it to room temp before you cover in fondant.

Certainly it can be baked, filled and crumb coated now Do not cover w/fondant yet.
Wrap it in plastic wrap or a plastic bag and freeze it. Then two days before wanting to finish it, take it out of the fzr (that night) and put it in the frig; If you have crumbed it, it can go directly from fzr to room temp UNwrapped. The crumbcoating will keep it fresh it will NOT dry out.

I COMPLETELY second the bake-now-freeze-for-later suggestion! As a matter of fact, moisture is locked into the cake SO well this way, that I refuse to bake right when I need my cakes, I ALWAYS make sure that my cakes get at least an overnight stay in the freezer. The texture is fabulous afterwards! The key, though, is to wrap very very well! Even if you don't crumbcoat prior to freezing. HTH!

I have been wondering about this...when you freeze cakes....is it better to thaw in fridge (with wrapping)? or thaw on counter (wrapped or not?)
TIA

It makes little diffrence if you thaw in frig or on counter. What is important is that the cake be covered - whether w/plastic wrap/bag or crumb coat.
If it is crumbed, then UNwrap - again, either on counter or in frig. Usually I like to take my cake out of the fzr the night before I plan to work on it and put it directly into the frig. The next day it is ready to work on

It won't be eaten, does that make a difference?

If it won't be eaten, I would definitely use a fake cake (styrofoam). There's no chance of the buttercream softening making the fondant sag. It's so much easier for display cakes. Transporting it is a breeze too. I hope that helps!