What EXACTLY does your lease say about late fees? Please quote the language.
Why would your landlord have to remind you about the provisions of a contract to which you are a party and of which you (presumably) have a copy?
So...you admittedly breached the contract by not paying rent when it was due. Right? Does the lease require that the landlord formally "charge" you in this regard?
Rather obviously, no one here is in any position to explain your landlord's reason(s) for doing or not doing things.
Depends on what the lease says.
Sure you can (but the ability to sue is obviously different from the ability to win).

