"If I apply keywords to the catalog files, will those keywords then be synced to the images on the hard drive when I reconnect the drive to Lightroom?"
"Are the keywords then embedded in the images so that if I create a new catalog on a second computer with the same images now copied from the external drive to that computer, when I create the catalog, will the keywords appear in the new catalog?"
No. All metadata including keywords is stored in the catalogue (a database) in which you entered it. However, if you want to embed or write the metadata into the images, you need to tell Lightroom to do it - use Metadata > Save Metadata. This includes all IPTC metadata including keywords, and adjustment metadata, but there are things it doesn't include (flags, stacks, history steps...).
But if you want to create a new catalogue on another computer, with the same images, the best way would be to copy the catalogue from the first computer or use options such as File > Export as Catalog which clones the catalogue.
For travelling, there is also the newer option of smart previews. It might be a bit much for you right now (or maybe not - there are plenty of questions more basic than the one you pose!), but I recently travelled and took on the laptop only my catalogue and smart previews for all photos (no standard or 1:1 previews). This meant I could add keywords and other metadata - but I could also make some adjustments to the images and email copies of them, etc. When I returned home, these adjustments would apply to the originals.
Thank you very much, John! That is exactly what I needed to know. Since the first part of the project is simply keywording, I might use the save function once it's completed. My advantage is the the folders in which they are placed are already named with most of the info I'd need.
As to the smart previews, I know that I can apply them selectively as I need them. This trip isn't going to be for that.
Another question becomes: If I have the identical images stored on another computer in exactly the same file structure, is it possible to import the catalog to link it to those photos? I have them stored there and in a third location as well.