The first question has a positive answer: if there is a first integral, the space where the movement occurs can be reduced by 1. The most prominent example is when there is energy conservation, on which the system is restricted to a 2N-1 space.
The general case (2N dimensions) is treated in the same manner (see e.g. goldstein's Classical Mechanics). However, it is not true that if you have M first integrals, you can reduce the system to a 2N-M space. The Poisson brackets between then must be zero in order to you be able to reduce the dimension. In this case, the constant of motions are said involution.
In case you find N constant of motions involution, the system is said integrable, and the movement is restricted to a N-dimensional torus.