Are there any tricks to making a light-textured "whipped cream cake"? During a web search, I found a cake recipe that uses heavy cream instead of butter. 

However, when I made it, the cake texture was terrible; thick and doughy even though I tested the cake with a toothpick and the toothpick came out clean.This is the recipe I used:

2 cups cake flour 
2 teaspoons baking powder
dash salt 
1 1/2 cups heavy cream 
1 1/3 cups white sugar 
2 eggs (room temperature) 
1 teaspoon vanilla extract


Preheat oven to 350 degrees. Grease 2 8" cake pans; line with parchment paper; grease again.

Sift cake flour, baking powder and salt in a medium bowl; set aside.
Using an electric mixer, whisk the cream with the sugar until stiff peaks form. Add the eggs, 1 at a time, whisking well after the first before adding the second. Stir in the vanilla. Fold in the flour mixture until just incorporated. Divide batter between the prepared cake pans. Bake 25 to 30 minutes, or until a toothpick inserted into the center of the cake comes out clean. Cool cakes in pans for 10 minutes, then invert on to cooling racks; re-invert so that the tops of the cakes are facing upwards. Cool completely.

I live in Boulder County, Colorado, so I decreased the sugar to 3/4 cups (I do this routinely, eve when I lived in southern Indiana, because I believe excessive amounts of sugar are used in American baked goods (leavened without yeast).

I WEIGH all ingredients and mixed the cake according to the directions; I folded the flour mixture just until there was no dry flour. What did I do wrong? Are there adjustments that should be made to the recipe, such as increasing the baking temperature? I understand that this type of cake should have a velvety, melt-in-your-mouth texture. How can I make this type of cake successfully?
If the directions say that you should beat your cream and sugar until you have peaks, then whisk in eggs then that may be a possible cause of your results. You've just put loads of air into the cream, then you're supposed to beat eggs into it, which will punch some of the air out. What I would do is beat the eggs separately (I'd not just beat them until mixed but beat them until they start to turn pale, that adds more air), then fold them and the flour into the cream. 

Be careful not to overbeat the cream as well, if you overshoot stiff peaks you'll start to turn it to butter, and then you've lost your lift. I'd stop a bit short of stiff peaks, leave it a bit looser. 

Leaving that much sugar out without reducing the amount of cream would mean that you've got too much liquid in your cake. If you have too much liquid it won't be able to crystallize and the cake, while it will rise, will not be able to hold shape and will collapse. I would compensate by leaving some of the cream out. If you weigh all the dry ingredients you can work out a ratio of dry to wet ingredients (the eggs and cream separately). Then if you weigh the amount of sugar you plan to remove you can work out how much of a percentage of each wet ingredient to take out. Or you could just wing it and use 1 1/4 cups cream and 1 3/4 eggs and see how it goes, that's probably about right anyway.