Of course you address this in your post (actually, everything i'm about to say you've already addressed) but I think for the purposes of the student it's just easier to learn how to "plug and chug" so to speak (i know...i know this doesn't actually facilitate learning the material, but if it helps understand how to at least attempt these problems, why not?)

Let us write all the things we know,

Our function is defined as,

$\displaystyle F = xy \hat i - x^2 \hat j + (x+z) \hat k $

Our surface is defined as,

$\displaystyle z = 6-2x-2y $

And for the computation of our integral,

$\displaystyle \hat N dS = +/- ( - \frac{ \partial S }{ \partial x } \hat i - \frac{ \partial S }{ \partial y } \hat j + k ) $

Notice how we have a plus and a minus, this means we need to find both of these to get to the total flux! If you just want upward or downward, only compute the flux integral for either of those.