Cool,
I don't think window screen will hurt or help anything. It will decrease circulation in the hive.
As far as SHB, they enter the hive from any opening, cracks, AO holes, uneven boxes, knot holes, etc. Keeping your hives really strong is the best deterrent for SHB.
IMO putting the robber screen on when you make the split is the best way to go. This time of year make your splits/nucs strong. I spent the weekend making splits, the week before was getting equipment ready which included robber screens on the nucs. (I elected to try eliminating the lower entrance of the nucs and drilled a 1" hole entrance when I ran out of robber screens).
The robber screens are also nice moving screens, smoke them in just before dark, close the entrance and make the move, then reopen the entrance. I use the nearest branch with a bungy cord to distract them when they exit the new location. If I'm in a hurry I'll lean a bottom board or 2x4 at the entrance to make them pause, (yesterday I used rope in front of the nucs with the 1" entrance). Moving the hives is more traumatic for the beekeep than the bees.
Robber screens will not prevent mice infestation, remove them late fall and put on your mouse screens or what ever you use for mice.
Time is an issue for me as I don't live near my hives. Fast easy and cheap is how I manage my hives. Like you as a beginner I purchased a lot of equipment to try it out, if it was worth duplicating I did so, if not I purchased or moved on.
The BeezNeedz product does seem superior to the Mann Lake product, when I get some spare change I'll pick up a few.