Pages is going to do this just fine. But key is putting it in Page Layout mode, an option in the File Menu. Now instead of working like a word processor, it works like DTP software.
Then you can drag these pages split in the wrong place, onto individual pages and link the splits pages back together again on single pages. For example if one your sheet music pages is split at 50%, you can pull that up to the upper half of page, put the lower 50% onto the bottom half of a page and now you have the page of sheet music reassembled on a normal page. Repeat for all of the "broken" pages you have.
You'll have to do this manually for each page, so it will be a pain if you have a LOT of pages. But it will also easily work and you can be very precise in lining them up. You can also "Edit Mask" to trim unwanted stuff off so that what you end up then printing (or "printing" to another PDF file from Pages) is only the sheet music on regular paper as you seem to want it.
You seem to have a problem in #4 that is resolved by "showing thumbnails" in Preview and then dragging each page thumbnail into Pages and putting it where you want it. Think of this like the thumbnail pages are puzzle pieces and you have to reassemble the puzzle... or- in this case- you are making a bunch of probably 2-piece puzzles.
Else any paint/image program that can import PDF files is going to work too, OR you can open this file in something like Preview, export the pages as .png or .tif files and then merge them back together as images instead of PDFs in any app that works with images.