Does every projection operator satisfy $\|Px\| \leq \|x\|\,$? It's well known that an orthogonal projection satisfies $\|Px\|\leq\|x\|$.
Does this property hold for any general projection operator $P$, which is defined by $P^2=P$?

The property does not hold.   For example, consider
$$
P = \pmatrix{1&0\\\alpha &0}
$$
for some $\alpha \neq 0$. Note that $P^2 = P$, but
$$
\left\|
\pmatrix{1&0\\\alpha &0} \pmatrix{1\\0}
\right\| = 
\left\| \pmatrix{1\\ \alpha} \right\| \geq
\left\| \pmatrix{1\\0} \right\|
$$
