Sure you can fill it. Look at YouTube for some tres leches videos. Just know that it's normal for some liquid to pool at the bottom of the cake. You could fill it with whipped cream or pastry cream and put some fresh fruit in between layers. For a tres leches I'd use SPS to stack.
What kind of powdered sugar are you using? I think that may be the culprit. I can always feel the powdered sugar crunch between my teeth, but if it's a recipe which uses a higher amount of powdered sugar, it may make it worse.
I use heavy cream in my recipe too, and I use a lot. I heat it up before I add it to the recipe because I think the heat breaks down the sugar a bit.
Generally, yes, you'd use a board under each tier cut to size. However, some people don't do that, especially if they're using fondant. You may see some people just spread royal icing in between each tier.
Yes. This is me. LOL. But I also just like more cake than filling I guess. I do tend to pile on the buttercream so like AZ said, I like to have a balance with everything. I do like having three layers of cake and two layers of filling, but I don't have three of each pan size so I almost never do cakes like that (only for family).