Here is what happens...
The source emits photons in the direction of the plate. The plate will block most of the incident photons, but some will pass through the narrow slits.
When a wave passes through a narrow gap in a barrier it spreads out beyond the gap in a circular pattern (narrow here means that the width of the gap is of the same order of magnitude as the wavelength of the wave). Where you have two adjacent gaps, the spreading wave from one gap overlaps the spreading wave from the other. Where the peaks and troughs of the wave from one gap coincide with the peaks and troughs of the wave from the other, the peaks and the troughs become higher and deeper- where the peaks from one wave coincide with the troughs of the other, the waves cancel each other out.
With light waves or photons, this effect produces regularly spread dark and light areas. If the two waves from the two gaps hit a screen you can see the dark and light areas as 'fringes'.
If the screen is flat, the fringes will appear as alternate light and dark lines running parallel to the lengths of the slits and about the same length as the slits.
If you change the shape of the screen, the behaviour of the light itself doesn't change- in 3d space it continues to have light and dark areas as before, but the screen now cuts through the space in a different way, so the pattern of the interference fringes shown on the screen will change. If the screen has spherical curvature, you will still see the fringes as lines, but they will be curved and their spacing will change.
If you increase the height of the slits you will just increase the height of the fringes.
If you decrease the height of the slits, you will shorten the fringes, unless you decrease it so much that the height approaches the width of the slit. At that point, the light leaving the slit will spread out vertically as well as horizontally. Rather than seeing vertical interference fringes you will see a pattern of light and dark spots.
So, in summary, the spreading light from the two openings interferes with itself, regardless of whether any screens are present, producing regions of increased and reduced intensity in a regular pattern. Introducing a screen simply allows the pattern to be viewed- the shape and orientation of the screen with determine how it costs through the regions of light and dark, and thus determines how the patterns appear on it.