Yet another double slit experiment If single particles shot one at a time at the double slit do exhibit wave-like behavior, then if we put another screen right between the slits and perpendicular to the first screen, won't we also see that some of the particles form a pattern on the second screen as well? (if this is silly please remove my question)

 A: Assuming you mean some special screen 2 that "lights up" when hit from either side, then no, there won't be a pattern, because the length (or rather:time) from hole A to some spot on the screen will be the same as the length/time from hole B. There will be no destructive interference "on" the screen 2.
A: The second screen makes it two separate Fraunhofer single slits with their independent diffraction pattern, if the slits are narrow enough. On the vertical screen there will be spots uncorrelated, the same as would be found on the second screen if one closed off sequentially each slit. 
If the second screen were made partially transparent the degree of survival of the two slit pattern would depend on the percentage of beam coherence lost. Any scattering disturbs coherence. This experiment has used semi transparent covers on a double slit experiment :

Although the electrons (which were shot one by one) could still pass through the filtered slit, the filter caused more of the electrons to undergo inelastic scattering rather than elastic scattering. As the physicists explained, an electron undergoing inelastic scattering is localized at the covered slit, and acts like a spherical wave after passing through the slit. In contrast, an electron passing through the unfiltered slit is more likely to undergo elastic scattering, and act like a cylindrical wave after passing through that slit. The spherical wave and cylindrical wave do not have any phase correlation, and so even if an electron passed through both slits, the two different waves that come out cannot create an interference pattern on the wall behind them.

Something similar would happen if the second screen were semitransparent.
