Manipulate is has one of two meanings depending on what context it is being used in;
when used with a person or group, for example "I manipulated them", it mean to control in a negative way like by deceit or withholding the truth.
when you use it with a object, for example "I manipulated the ball.", it means you physically manipulated the object. This can be anything from picking it up, to turning it around or something like that.
To "deal with" can mean different things. A "deal" is a sort of agreement that is usually good for at least one of the people making the agreement. Usually to deal with means to make deals with someone, for example one company makes deals with another. or it can be used in the context of a problem. If you say, deal with a problem, you are finding a solution to the problem.
I hope this answers your question.