It depends on the cleanser. AHA/BHA are chemical exfoliants. They help dislodge dead cells that are stuck in your skin through chemical action. The cells will fall off through normal mechanisms: shedding, coming off when rubbing, etc.
Some cleansers are physically exfoliating meaning they have abrasives in them which help to physically remove the dead skin cells.
Cleansers can also have chemical exfoliants in them. However because chemical exfoliants work by longer exposure, it's better to keep them on by using a toner rather than a short exposure time with a face wash that you rinse away.
No, I only use CeraVe AM as my daily sunscreen because I am inside all day. Biore watery essence is saved for outside time, also because it's expensive and hard for me to source.