Lens manufacturers say that anything other than a microfiber cloth will damage the coatings on a lens. But microfiber cloths eventually pick up crap and they look like they’re a pain to wash.
How do you keep your lenses clean? Are you using cloth/kleenex or microfiber cloths? If you use microfiber cloths, how do you clean those?
Not if it’s moisturizing soap. The moisturizers are designed to remain after rinsing, which will leave an oily film on your glasses. Dish soap is designed to rinse clean, so you should use that instead.