Information from a characteristic polynomial 
(III) I know that since the highest multiplicity is 4, that the largest possible dimension is 4.
(IV) I believe that there are 3 distinct eigenvalues, but because of the multiplicities, there are exactly 7 eigenvalues?
I don't know how to tell if it's invertible, or the size of the matrix from the information given.

(I): $A$ can't be invertible because $0$ is an eigenvalue.
(II): $A$ must be $7 \times 7$ because its characteristic polynomial has degree $7$.
(III): Your answer is correct
(IV): I don't think that they mean to count eigenvalues up to multiplicity.  Yes, $A$ has exactly $3$ distinct eigenvalues.
