Hi Jason,
Welcome to the forum

Some of the items you list are definitely wishlist items. Others can be done.
I don't have time to address the full length of your post but I'll start with #1. Other forum members will, most likely, jump in and pick up from where I left off. If they don't, I'll come back to your post as time becomes available to me.
1) When I highlight - or try to highlight a few chords for cut/paste - highlighting doesn't always seem to work. Case in point: I want to highlight one measure to cut/paste -- can't do it. Seems two measures are required. The workaround is to highlight two measures then scroll back to the first measure. Before I found that this worked, I had keyed in manually many measures (with multiple changes per measure).
BIAB works on the principle of a 'cell' and the cell contains a maximum of four locations for the placement of chords. From a musical perspective, it's easiest to think of this cell as "one bar".
As you note, if one wants to highlight a single cell, it's necessary to drag the mouse across the two cells and, keeping the mouse button down, drag the mouse back. A single cell is then selected. This is normal behaviour.
I can see the value of being able to copy/paste the individual chords within a cell. That is a 'wishlist' item (the Wishlist forum is found
here).
In the meantime, deleting unwanted chords in a copy/paste is simply a matter of using the 'delete' button the keyboard. It's very quick.
2) I looked up how to place "NC" (rests). There was a previous topic very long ago covering this. The way this has been implemented since makes some folks happy. Maybe I'm biased, but it seemed that use of "NC" as a chord symbol was favored as a suggestion to add. How one uses NC is up to them. A red chord symbol which denotes a rest looks wrong for my purposes. Just to check - I saw the printed lead sheet (print preview) doesn't handle these rests in what I would consider a correct way. There's no indication either with a rest symbol, "NC", etc. in the printed version of the leadsheet either.
I'm not too concerned about the lead sheet as I would generally use other software to generate that. However, in practice - I forgo use of a lead sheet and read the transposed "what's on the screen" (chord sheet view). This is what is fueling the suggestion that the current implementation is perhaps a "bug" at least to my understanding of notation.
In relation to the Lead Sheet, it's possible to add text boxes to this. Such additions are accomplished through the Notation Window.
Go to the "Help" menu and search for "Section Text". You'll then be guided to information relating to these text boxes.
Being able to include "NC" on the chord sheet view is a Wishlist item.
(It's worth noting that the Chord Sheet is where BIAB gets its instructions for generating a backing. In this regard, the Chord Sheet's purpose is to program BIAB's accompaniment. This is why sometimes chords are red and periods and carets are used to accomplish specific behaviours. This is the programming language of BIAB/RB. The Lead Sheet view is what is intended for the musician to follow.)
Regards,
Noel