Yes, I would remove makeup before washing face. I use the makeup removing wipes from Exact brand. Only thing that doesn't irritate my skin or cause breakouts.

I always remove my make up before washing the skin(to clear it not only from make up but from dust and other factors too). I would say that it is individual, cause I have quite oily skin, and I've tried so many products (too many to count) until I've found the one that works for me. I can't recommend it to you, cause I am not sure, whether, it is doing to work for you or not you'd better consult with a good beautician.

Always remove makeup before doing anything to your skin. If you currently have a skin care routine, do what you would normally do. You can gently wipe the makeup off with a washcloth or shammy, then use the cleanser that comes with the regimen. I'd definitely use a product specifically for removing eye makeup, since the skin around the eyes is much more delicate than the rest of your face.

Any moisturizer could be the best remover.