I think health food stores will probably be your friend here since they have some more options. Asian stores could be good, too, since there are so many rice products. There is tapioca pudding makes a good dessert. Kozy shack makes a pretty good one, and it’s actually in mainstream grocery stores, too. They also have rice pudding and chocolate pudding, neither of which have wheat or corn in them. There is ice cream, too. Most Haagen-daaz doesn’t have wheat or corn. Not sure about cake mixes. That may be harder. But you could probably make a cake without much more effort than a mix just using rice or oat flour. Cake mixes are usually just flour, baking powder, salt, sugar and flavoring, so the only extra step going from scratch would be adding those by hand.
For pasta, there are lots of options. Brown rice (Tinkyada is a good brand), buckwheat, soba noodles. There are tons of condiments without wheat or corn. Red boat fish sauce is awesome, and has neither. Same with Coconut Secret coconut aminos. The combo of the two is great on rice (very similar taste to soy sauce, which often has wheat). Organic ketchup should not have corn. Annie’s brand uses cane sugar, not corn syrup. Hope this helps!