Is it possible to make a selection in Microsoft Paint beyond the current screen resolution?

While selecting an area by holding down the left mouse button and dragging, you can use your mouse scroll wheel to scroll up and down the image. This allows using a selection larger than the screen resolution in the vertical direction.

However, you cannot scroll left to right and would probably need to zoom in/out for this.

In my experience, not directly. However, there's still the workaround of saving a copy of the image and cropping the copy to the size you want, then Ctrl + A , Ctrl + C, Alt + Tab, and Ctrl + V. ;-)

Fwiw, newer versions of Paint include the option of zooming out, which should help in your case, but I don't know if I should recommend you to upgrade. Give it a try anyway, if you can, and decide for yourself.