Finale 2010 was released in summer 2009, 11 years ago.

I am not sure I remember, how things were done in Finale 2010.

The following is a “shot in the dark”.

… Even after following the directions given using Staff List, Category Designer, things I put onto a multiple 19-20 stave Brass Band score are not attached when I try to edit or extract parts …

1) You probably do not have to extract parts anymore.

Instead, use the linked parts inside the score document.

2) Much has changed since Finale 2002.

a) Staff Lists are used in the Repeat Tool.

In the Expression Tool you have instead two other list types:

Staff Assignment Lists

and

Score Lists

b) There are two types of expression categories

I. Categories that use Staff Assignment Lists (= Dynamics, Expressive Text, Technique Text)

II. Categories that use Score Lists (= Tempo Marks, Tempo Alterations, Rehearsal Marks)

You can create your own, custom expression categories, by duplicating existing categories.

To create a new category of type I, duplicate one of the categories Dynamics, Expressive Text or Technique Text.

To help better we need you to be more specific.

What expressions are not shown in the linked parts?

… when I extract parts or try to edit, the staff name does not appear, which on 2002, was done automatically …

My first guess would be that you should look in the Staff Attributes.

There are options to show Staff Names, in the Score, and in the Part.