When I add a link to my link directory, does that link always remain on the same page, accessible at the same URL?

This depends on your Link Page Format settings. There are several different combinations of settings that will force each link to always remain on the same page:

1) If under Link Page Format, your Layout Style (or under Advanced Options, the Link Page Division) is set to one of the Single Page options, then all of your links will always remain on the same page because there is only one link page.

2) If under Link Page Format, Advanced Options, you have Link Page Division set to Separate Page For Each Subcategory, and your Link Order is set to Date Added (oldest first).

3) If under Link Page Format, Advanced Options, you have Link Page Division set to Separate Page For Each Category, and your list of categories under Link Categories does not include any subcategories at all, and your Link Order is set to Date Added (oldest first).

4) If your Number of Links Per Page is set higher than the number of links in any category (if you have a separate page for each category) or subcategory (if you have a separate page for each subcategory), so that no category or subcategory need be split into multiple pages.