FIR Filter - Transfer Function
HINTS:


Just multiply the terms to get $H(z)$ in the form $$H(z)=h[0]+h[1]z^{-1}+h[2]z^{-2}$$ If you know the four types then you should immediately see which type it is.
Your frequency response looks good. The term in parentheses is real-valued. Does it change sign? You can directly write down magnitude and phase from that representation (just watch out for a negative sign, and what that means for the phase).