On the 1863 Nashville cover:
Yes, Union occupancy of Confederate territory. Also,
Letters could still be sent collect during this time (obviously?).
Well, not quite. A simple overview is that prepayment became mandatory in the US in 1855. The Nashville cover falls under the postal act of Mar 3, 1863, which rates underpaid/unpaid mail at double the underpayments, thus 2 x 0.03 = 0.06. I would call this non-payment as tolerated, but penalized to discourage the practice. Others may have a better phrasing.
Somewhat related to that are the unpaid covers mailed during the civil war which are certified with a signature/title of an officer or chaplain as sent by a non-commissioned soldier/sailor and allowed to pass through the mails collect to the destination at only the single-due amount, authorized by the same act noted above, section 27. Thus the "due 3" ratings on a lot of civil war soldier mail. Commissioned officers did not get this mailing right, thus having to prepay or have their recipients pay double-due, which may be what your due 6 cover represents!
(I used the recent reprint of "Appleton's United States Postal Guide, 1863" which contains the 1863 act.)