The 7th Armoured Division, aka "Desert Rats", wore a patch that was a BLACK square with a RED rat with a long tail. There was also a Brigade that wore a similar patch with a GREEN desert rat image. The square RED patch with a silouette of a BLACK cat is the patch for the 56th 'North London' Infantry Division. I think this is the patch that you refer to. For an example of this patch, see this link. http://www.custermen.com/ItalyWW2/ArmyOrg/BritishOrg.htm
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One question at a time. That makes it easier to search for it again. The Swastika was a symbol adopted by the Nazi party and Hitler made it a national symbol for Germany. It was worn by every military and government organization in some place on their uniform or hat. It was used in everyway; on official stamps, money and every military medal. Some had a small insiginia and you have to look closely to see it but it was used in many ways.
It doesn't "mean" anything, it's just a symbol chosen to be visually distinctive.
Black was somewhat a national colour for Prussia. Their symbol was the black eagle and their uniforms sported black as well. Their battle standards were probably black, so raise high the black flags probably means that they should raise their battle banners and flags. Armies in those days carried a lot of flags as also seen in the movie.
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Red symbol of burning flame, and arouse the enthusiasm of the people for free.