If group delay is a constant for all $\omega$, Does this system have a linear phase response?
This is a homework type question, so I will only give hints and no solutions. You should know that the group delay is the negative derivative of the phase (with respect to frequency), so if the phase is a linear function of frequency, the group delay must be a constant. This should answer your first question.

Concerning your second question, why don't you just compute the phase of the given frequency response? It may be helpful to rewrite $H(e^{j\omega})$ as

$$H(e^{j\omega})=2e^{-2j\omega}\left(1-\frac{e^{j\omega}+e^{-j\omega}}{2}\right)\tag{1}$$

and to realize that the term in parentheses is real-valued.

EDIT:

The frequency response $H(e^{j\omega})$ can be written as

$$H(e^{j\omega})=M(\omega)e^{j\phi(\omega)}\tag{2}$$

where $M(\omega)$ is the magnitude, and $\phi(\omega)$ is the phase. Now compare $(1)$ and $(2)$ and figure out what $\phi(\omega)$ must be.