MacOS – Add Cut file/folder option to right click or Context menu

Keyboard method: Cmd-C then Opt-Cmd-V does the cut&paste for files on Mac.

Mouse method: Drag the file from one folder to the parent of the target folder (ie, if moving to Documents:Financial, drag to Documents). Hover on the parent folder for a few seconds, and it will spring open. Then you can continue dragging the file to the target folder. (note, the mouse method may result in very long hover times, if you're dragging a huge number of files, eg 1,000 files)

Menu method: It's not part of the Apple menu system to 'cut' files. The menu Cut option is grayed out, and becomes enabled when text is selected. But not files. Here is an in-depth discussion on Apple's discussion forum.

AFAIK no, but you can make an new Service with the Automator.app.

And the new service will appear in the "Services" submenu in the context menu.

Assuming than you're familiar with scripting and Automator - if not, enter more details what you want achieve with the URL... (for an example can check: https://apple.stackexchange.com/a/63362/6838)