Taxonomy terms won't arrange themselves by ID but will work alphabetically instead I added a large number of taxonomy terms in the order I want. Their term IDs reflect they are added in the correct order. However, everywhere they are displayed alphabetically. Particularly, inside a hierarchical select widget they are displayed alphabetically - when I see their IDs (856, 847, 843 - not a correct numeric order but correct alphabetic A, B, C) they check out fine so they were added properly. How can I control this?
The order in which you enter the taxonomy terms is not important to how they are displayed. Drupal defaults to alphabetic ordering, but you can adjust this order manually.

Go to Content management > Taxonomy > Your vocabulary: List terms (via admin/content/taxonomy/list) and reorder the terms by dragging the handle on the left. This is also the order Hierarchical Select uses.

You can also try out Taxonomy Manager to see if this provides you with a better interface for reordering large numbers of terms.