1) Copy or move the catalog file, i.e. the file with the extension < .lrcat> and possibly the previews, i.e. the folder with the extension < .lrdat> to the external drive.
I would recommend creating a top-level folder on the external named "Lightroom catalog and previews" and put catalog and previews in there.
Note: You can put the catalog wherever you wish; it doesn't have to be in the default location. And, the catalog does not have to be on the same folder as the photos.
Then - on both computers - double-click the catalog file and Lr will open with it. Lr will "remember" the location of the catalog. Thereafter by default, Lightroom opens the most current catalog at startup.
2) Yes, but make sure that the drive has the same drive letter on both computers. If the drive letters are different you will get the "missing image / folder" message.