Struggling with uncompensated/compensated demand
First, we want to find the optimal good baskets, ie how much to buy of goods $x$,$y$ and $z$ to get the maximum value of U out of it.

Let's write down the $MRS$ for $x$ and $y$ against $z$. For this we need to differentiate the utility function for each of the variables:
$d_xU(x,y,z)=a/x$ , $d_yU(x,y,z)=b/y$,$d_zU(z,y,z)=1$. We hence deduce the rates of substitution to be $-1a/x$ and $-b/y$ respectively.

So we have:
$p_x/p_z=a/x$ and $p_y/p_z=b/y$

The budget constraint is $p_xx+p_yy+p_zz=I$ which we rewrite
$p_x(ap_z/p_x)+p_y(bp_z/p_y)+p_zz=I$, which simplfies in
$ap_z+bp_z+zp_z=I$

$z^*=(I/p_z)-a-b$

$(x^*,y^*,z^*)=(ap_z/p_x, bp_z/p_y,(I/p_z)-a-b)$

Good news, you were right :).

The cross price effect is the rise of demand in good $x$ following a rise in the price of good $z$. Just calculate the difference between $x^*$ for $p_z$ and $x^*$ for $p_z'=p_z+\epsilon$ (using $x^*=ap_z/p_x$).

Finally, the six curves asked are how demand change when price change holding income ($I$) constant, or utility ($U$) constant. Draw same by taking two prices fixed for each situation.