Does aspartame cause side effects like insomnia, anxiety, or breathing problems?

Apparently so. As dr. Howell says, there's controversy about this, as there's money in selling aspartame-sweetened diet drinks and more. However, cleansing your system of toxins when you have unusual symptoms makes sense. "non-nutritional" sweetener is what it's euphemistically called.
Possible. There is a lot of controversy about aspartame - whether it is safe or unsafe. There are arguments on both sides. High quantities of aspartame could cause side effects in some individuals.Too much of one thing is usually not good idea. If possible, it is best eat/drink a variety of foods that are natural and not processed. If you need a sweetener, try something like stevia or truvia.