Take a look in

Preferences - Edit

Is the option “Respect Instrument Ranges when Copying or Changing Instruments” selected or de-selected?

It is selected. When it's turned off, the trombone part is one octave too high. 

Just discovered that Trumpet 2 WILL transpose to trombone correctly, but not Trumpet 1. Both trumpets are set up the same in the score manager. Weird?

… While trying to copy/paste trumpet to trombone unison pitches

the trombone appears one octave too low …

… It is selected. When it's turned off, the trombone part is one octave too high …

Let me get this straight:

You are copying Trumpet 1 to the Trombone - right?

And the pitches are supposed to be unison - right?

When “Respect Instrument Ranges when Copying or Changing Instruments” is on, the Trombone is one octave lower than Trumpet 1, in both Layout and Playback - right?

When “Respect Instrument Ranges when Copying or Changing Instruments” is off, The trombone is one octave higher than Trumpet 1, in both Layout and Playback - right?

Correct. Here is a screen shot. It almost seems like the F or F# in the trumpet part is affecting the transposition.

By you word “Correct” you mean both Layout and Playback - right?

Obviously I can not hear the Playback, but I can see the Layout.

Only one measure is selected.

Since only one measure is selected, I suppose that only the selected measure was copied - right?

1) In the selected measure I can see that Trumpet 1 and Trombone are playing unison pitches.

That is what you need - right? (even if it is rather high for a Trombone)

2) On The Other Hand, in the three measure before the selected measure I see a different Layout:

In the first measure the Trombone is playing two octaves lower than Trumpet 1.

In the second measure the Trombone is playing one octave lower than Trumpet 1.

In the third measure the Trombone is playing two octaves lower than Trumpet 1, just like the first measure.

Sorry, yes to both layout and playback.

In the screen shot, measure 2 and 4 transposed correctly, but not 1 & 3. 

… In the screen shot, measure 2 and 4 transposed correctly, but not 1 & 3 …

If I understand you correctly, you copied all the four measures (even if only measure 4 is selected) - right?

Sorry for being dense.

You tell that “measure 2 and 4 transposed correctly”.

But only measure 4 has Trumpet 1 and Trombone playing unison pitches.

In measure 2 the Trombone is playing one octave lower than Trumpet 1.

How can measure 2 be “transposed correctly” when the Trombone is playing one octave lower than Trumpet 1 ???

As I see the Layout, only measure 4 shows unison pitches.

I suggest you turn off this "transposition while copying music" feature. Now copy the trumpet to the trombone staff. Now you go through the trombone part and put it in the octave that you want. Finale is not a human brain and the decision about what ovtave to use should be a human decision.