$P$ is projection matrix iff $A$ is reflection matrix? I have the following definition

An $n \times n$ matrix $A$ is a reflection matrix if and only if $A^2 = I$ and $A^T= A$. A projection matrix is $P = 1/2(A+I)$.

I was wondering if I can conclude that $P$ is a projection matrix if and only if $A$ is a reflection matrix. If it can be said can you please explain why?
Assuming that this is true can I say that $A^2 = I$ and $A^T = A$  if and only if $P^2 = P$ and $P^T=P$? 

No, a projection matrix $P$ satisfies $P=P^2$, and is not symmetric in general. Only orthogonal projection matrices are symmetric. Furthermore $P^2=I$ would imply $P=I$. So symmetric involutory matrices $A$ with $A^2=I$ are something different than projection matrices.
Edit: You have added the relation $P=(A+I)/2$. Then of course we have
$$
P^2=(A+I)^2/4=(A^2+2A+I)/4=(I+A)/2=P.
$$
