Can my landlord come on the property any time he wants?
Each state has its own law on what is and isn't reasonable. The lease is only a red herring because if the lease exceeds the law then the lease is not valid.
Check the local law for your state under renters rights. Google the following
[State name] renters rights landlord entry.
On the property, yes - in the house, not unless it's an emergency...any other time he needs to give you notification that he is entering your home.
Check your lease. Some landlords put it in the lease that they have the right to go inside if they want. Every place is differant.
yes as long as it is reasonable. if he need to check something out he can.