If I make cover songs, can I remove a verse and move sections around?

You would be creating a new arrangement and that may or may not be considered a derivative work. The first point in know if something is a derivative work is whether the original songwriters can sue you for infringing on their copyright. With such a cover song, there is less likelihood that they will sue or demand removal of the song from its sales sources, because the original songwriter will be making money for each copy you make. The second issue is whether you will be able to claim your own copyright on the arrangement. The original songwriters are less likely to want you to do that and more likely to sue to stop you. You need their permission to create a derivative work on which you claim copyright. To get really good advice and legal help, take your song to one of us music lawyers and let us hear just what you are planning. A lawyer can also help you contact the publisher to either get permission or find out the publisher does not want you to change the song in any way. Knowing this in advance can save you a lot of money and trouble.

Yes, you should be able to do this.

As long as you have the license to publish derivative works, you will be fine.

A cover song, no matter how much you change it, is a derivative work. Consequently, it requires an assignment or license to not be infringing.