Confused about shifting convolutions
Whenever you have doubts regarding the properties of the convolution operator, you should resort to its definition.
Let $x_s[n] = x[n+2]$ and $h_s[n] = h[n+2]$. Then:
$$y_4[n] = x_s[n] * h_s[n] = \sum_{k=-\infty}^\infty x_s[k]h_s[n-k]$$
If we replace with the definitions of our discrete functions:
$$y_4[n] = \sum_{k=-\infty}^\infty x[k+2]h[n-k+2]$$
See what happens if we make the change of variables $m = k+2$:
$$y_4[n] = \sum_{m=-\infty}^\infty x[m]h[n-(m-2)+2]=\sum_{m=-\infty}^\infty x[m]h[n+4-m]$$
Does that final expression sound familiar? You were right indeed in your OP, as that is exactly what you thought:
$$y_4[n] = \sum_{m=-\infty}^\infty x[m]h[n+4-m] = y_1[n+4]$$