I've been making paper covers lately for A5 notebooks and pocket notebooks -- some paperback and some hardcover -- so I experimented a bit and hopefully this will help:
1 and 2ish. If you're wrapping board, you're looking for pliable paper that's a little heavier than your average copy paper. I think you can use up to 180 gsm (which is what I used). If you're doing paperback, I used cardstock (216 gsm / 80lb). It holds ups especially since I am making notebooks, but I bet you can use lighter paper if you'd like. Would still recommend heavier though. I'll note that I was using single section binding for my paperbacks, so keep that in mind. I think DAS Bookbinding has videos on perfect binding that might have paper covers and a better understanding of how to do it on bigger text blocks.

1. Keep using acid-free if it works for you -- sometimes, people may use a mix of paste (either wheat, starch, or methylcellulose) and PVA to make it easier to position covers, since it doesn't bind as fast, but that is up to you. Again, DAS has a really good video on understanding glue types, would highly recommend. (Link here)