Wolfram|Alpha can answer all of your questions, I think.
Picture of $\vec F_1$
vector plot (sin(x),sin(y)) for x from 0 to pi and y from 0 to pi
Picture of $\vec F_2$
vector plot (sin(1-x),sin(y)) for x from 0 to pi and y from 0 to pi
(Are you sure you didn't want $\sin(\pi-x)$ instead of $\sin(1-x)$?)
Picture of $\vec F_3=\vec F_1+\vec F_2$
vectorplot (sin(x),sin(y))+(sin(1-x),sin(y)) for x from 0 to pi and y from 0 to pi
Picture of $\vec F_4=\vec F_1\times\vec F_2$
For this we have to be careful. The main definition of cross product does not apply to 2D vectors like these. However, if we treat $\vec F_1$ and $\vec F_2$ as 3D vectors with third component $0$, then we can take their cross product as normal, and it's somewhat common to do so. We'd get $\left(0,0,\sin(x)\sin(y)-\sin(1-x)\sin(y)\right)$. Since all of the values of the cross product are multiples of $(0,0,1)$, we may as well just take the number in the third component, which is occasionally called the "scalar cross product" of 2D vectors. This is Wolfram|Alpha's default interpretation, as in their vector algebra examples.
With this in mind, the plot will just be of that scalar function (you could pretend there are "up" and down arrows from the $xy$ plane to the plot, if you prefer), and we get:
vectorplot cross((sin(x),sin(y)),(sin(1-x),sin(y))) for x from 0 to pi and y from 0 to pi