I leave my pizza stone in the oven all the time. It's in the middle rack, which is where I put most of my cakes/muffins I make (unless the recipe states differently).
Side note: I've noticed if my cake pan isn't completely on the stone (if it's hanging off), somehow the part hanging off doesn't rise like the rest. Weird. Not sure why this is, or if it's just my imagination.
Can you post your recipe? It very well could be you're beating the eggs too much; most carrot cake recipes I know of just call for mixing the ingredients together, not beating the eggs like you would in a 'regular' cake recipe.
It also could be your leavening; old baking powder?
I didn't get to finish and accidentally posted. continued:
2 tsp baking powder
3 c grated carrots
1 c chopped pecans (a must)
Grease and flour 2 9" round cake pans. Preheat oven 350*. Sift dry ingredients twice & add nuts & set aside. In a large bowl beat oil and sugar well. Add eggs & beat well. Add four mixture and blend. Fold in carrots small amounts at a time. Bake for 30 mins. I have had to bake this cake as long as 45 mins. depending on the oven.
Oh and of course it has to have the Cream Cheese frosting.
1 8oz pkg cream cheese
1/2 to 1 stick of butter room temp.
4 cups of confectioner's sugar
2 tsp vanilla
Blend with an electric mixer and spread on cake.