Thank you for the description, that's really helpful.
Would you try a slightly different flow? I want to see if I can pinpoint the issue by having you try to add tags/categories in a different manner.
Go to Posts > All Posts
Select the checkboxes for those first three posts
From the Bulk Actions drop-down menu above the top post, select "Edit" and click "Apply"
Check the categories you'd like to add, and enter the tags you'd like to apply
Click the blue "Update" button
Let me know the result!
(This process assumes that all three posts will have the same categories and tags - if this isn't the case, you can instead select them one at a time in the same way).