Just to note: There will not be a delete option on a category on the Category page if posts are assigned to that category. (Added see below.)
If you want to delete a category, you have to first change those posts to another category. You can do this on the Posts page, sort by category, then a Quick Edit to change the category.
Once there are categories without posts, then you can delete that category.
Changing the values in the table may cause other problems. Probably not severe, but I would guess that you have many posts without categories because you deleted those items in the table.
Added
Apparently the current version of WP allows you to delete categories that have posts assigned.
I recall that not being the case in prior version – 4.9-ish – but cannot tell the exact version. It was within the last year, though.
Answer modified with this new information.