I recently baked a lemon mascarpone cake (recipe from lovelifeandsugar). I tweaked the recipe a bit by using yoghurt instead of sour cream (because I didn't have any), used cake flour instead of AP, and used two tablespoons of oil instead of water. I also whipped my egg whites separately before folding them in. The cake turned out really tasty and moist except... when I tried to frost it it crumbled away. I'm wondering how do people get such study cakes like what you see in videos and photos and what causes the structure to be so strong? I tried to mix as little as possible after I added the flour ( I did it by hand after I added the dry portion in). I did put it in the fridge, but it didn't help much. Any suggestions will be greatly appreciated.

One other thing I found that was really strange was that my cupcakes turned out fine. I had leftover batter and I baked them into cupcakes. They didn't crumble when I frosted them with a piping tip, nor when I frosted with a small spatula. They were a bit too brown on the outside though, so if have any idea on how to fix that it would also be helpful.

It sounds to be like you may have over tweaked the recipe.  Cake flour and AP are not interchangeable, measure for measure.  A recipe that calls for All Purpose would require a larger volume of cake flour.  Why substitute oil for the water, since the recipe has 3/4 cup butter?  Why beat the egg whites and fold them in instead of adding them as the recipe calls for?  Beating the egg whites would tend to make the cake more tender.

Not all cake recipes are sturdy.  I make all kinds of cakes from scratch and many of them are just not sturdy enough for a reliable stacked cake.  My go to recipe, especially for carving/sculpting, is kakeladi's *Original* WASC recipe.  Pound cakes are also very sturdy and close textured.                                           https://www.cakecentral.com/recipe/7445/the-original-wasc-cake-recipe 

Thanks Sandra for pointing her to my recipe :)  My only other thought is that maybe she is using the wrong type icing recipe or the wrong consistency.  Here is a great one offering both consistencies:  https://www.cakecentral.com/recipe/22469/2-icing