Drawing conclusions from impulse response of a discrete LTI system
You're right about your conclusions. And now a few hints to help you solve the questions:

Your textbook should state somewhere the definition of even and odd parts of a (real-valued) sequence: $$h_e[n]=\frac12\big(h[n]+h[-n]\big)\\h_o[n]=\frac12\big(h[n]-h[-n]\big)$$

In this case (FIR filter), the difference equation is simply given by the convolution of the input sequence $x[n]$ with the impulse response $h[n]$: $$y[n]=\sum_kx[k]h[n-k]$$