In the first case ($\sigma / \varepsilon_0$), if you extend the gaussian surface of the cylinder to very far away from the surface of the conductor a central point will not be true in the calculation: The flux on this lateral surface will not be zero.
This is a central point in the case of a conductor: it is reasonable suppose that there are no tangential electric field just upside the conductor surface, or else there will be currents. So the point is that the tangential component of the field must be zero, adding no flux to the lateral area of the cylindrical gaussian surface...
The second case is trickier. It's totally non-intuitive to obtain as a result of a charge distribution a constant field as far as in the flux expression you have an electric field that should decay as $1/ r^2$ as you pointed.
I would pinpoint some arguments that I think can give you some hints.
(i) The situation of an infinite charged plane is quite artificial. It will off course never occur in a laboratory. That said, the calculation of a infinite plane is usefull if you think of a finite plane, for which you want to identify the field in the central regions.
(ii) This situation is a very good approximation for parallel plate capacitor, if the distance between the plates is very small and the deviation of the field from that occurs more noticeably in the ends of the plates.
(iii) About the dilema of a non-intuitive result I can think of a physical argument, why it must occur, which is current in the textbooks... Take a central point $P$ that is distant a quantity $x$ from the plane. 531
If you take for instance a small $dQ$ in the plane distant $x\sqrt{2}$ of this point, you will have a right triangle, $x,x,x\sqrt{2}$. Now, take the opposed $dQ$ that forms the same right triangle and sum the electric field of both charges in $P$. You will get an electric field perpendicular to the plane as a result that depends on the sine of the distance $dQ$-$P$ and the plane. If you take another point distant $2x$ first as the distance increase by 2, the field diminishes by 4. But this sine term increases if you go from $x$ to $2x$. So, all in all if you integrate over all the sheet the influences equalize each other... (I will try to develop a little bit better point (iii) soon). 586