Edit pdf files

If the page numbers all appear at the same spot on the page, and don't overlap with anything else, you can use the Redaction tool to do it. Create the redaction on the first page, then right-click it and there should be an option to duplicate it to all other pages, I believe. If it's not, you can use this (paid-for) tool I've developed to do it:

http://try67.blogspot.com/2009/07/acrobat-duplicate-annotations-to.html

Try the redaction tool of Acrobat Pro.

That is not really possible. 

You could use some scripting approach if there was a way to identify the 'part' to delete with a unique name that Acrobat would recognise. The best way is to go back to the original and re-export it.

Depending on its location, you could consider cropping the page, essentially cutting the top of the page where the page number appears. Use the crop tool, apply to all pages, but you might erase elements you want to keep.