Deriving of magnitude, phase response and impulse response of transfer function
HINT: The frequency response has the form
$$H(e^{j\omega})=a+be^{-j\omega}+ce^{-2j\omega}+be^{-3j\omega}+ae^{-4j\omega}\tag{1}$$
which can be rewritten as
$$H(e^{j\omega})=e^{-2j\omega}\big[ae^{2j\omega}+be^{j\omega}+c+be^{-j\omega}+ae^{-2j\omega}\big]\tag{2}$$
Now note that the term in brackets is purely real-valued. I trust that you can take it from here.