@Ynnad - here are some partial answers/opinions I received for some the questions you raised. A lot of this, as you may imagine, is speculation. My friend had replied a while back, but I've had a lot of things going on and I'm only now getting around to putting this together.
RE: "Paid according to number"
I really don't know this answer specifically. But, from logic and my understanding of how the redemptions were usually handled, Arkansas Treasury Warrants were numbered based on the date they were issued. For example: I have four different $10 ATW. The earliest dated one, March, 1862, has a serial number (left # on face) of 43,845, the second one is dated August, 1863 and numbered 100,396, the third one is dated May, 1864 and numbered 126,994, and the fourth one is dated March, 1865 and is numbered 189,581. So, I'm presuming that they would be redeemed according to first to last.
RE: "On War Bond" reference on the PCGS label
The "ON WAR BOND" reference has to do with printing on the face of the note or (in case of your note) it is NOT printed on your note (it was removed). There were generally three references printed or NOT printed on each side of the portrait on the note. 1) "ON WAR BOND" 2) "ON AUDITORS WARANT" & 3) these words were removed on the note.
For Example:
ON WAR BOND

On Auditors Warrant

Words Removed from these notes

RE: Two Different numbers on the notes
These numbers are generally serial numbers based on when the notes were issued. The PCGS label calls them "S/N" or "serial numbers" . I'm pretty sure the number on left is based on when it is issued (see first answer above). I'm not really sure what exactly the number on the right side represents.
Your note has the PCGS label with "Serial# 313-87224" referenced and the note has written numbers left - 87224 and 313 on the right. So, they are calling it a "serial number". I have another note graded by PMG where the 2 numbers are labeled "S/N 187581/59076". Here's what it looks like:

RE: What Dollar mean on CSA notes
Generally, a dollar in the North as the same dollar as it the South, except they were backed by different economic systems. No government in a war really wants to pay out specie (metals) for paper money (normal redemption in 19th Century). I did find a little bit of information from one reference book I have on Confederate states' money: "Confederate States Paper Money" by Arlie Slabaugh, 10th edition, 2000.

RE: What were Ark Treasury Warrants worth 1863, 4, and 5:
Generally, Confederate money was worth more in the beginning of the early than at the end of the war. Although not backed by actual deposits of gold and silver in the treasury, Confederate money was generally used in and accepted by most citizens of the South. Confederate money was originally intended to be backed by cotton. The South intended to control the value of the currency by its stockpile of cotton. I'll include another quote from Slabaugh's book.

RE: How many Arkansas Treasury Warrants were actually redeemed by State.
According to Slabaugh, the each of the Confederate state governments printed their own currencies. The Confederate government based its own existence on the idea that each state could follow its own course, ie., state's rights concept. He says Arkansas was one of the most liberal states in printing money. He doesn't really address the issue of redeeming the money of the states that they printed.
I know Arkansas printed and issued Treasury Warrants for the entire war. When Little Rock was occupied, the Arkansas Confederate state government continued to function in exile in Washington, Arkansas. I have been told by some of employees at Old Washington State Park that they still have in their archives the original engraving plates used to print Arkansas Treasury Warrants. I have a couple of examples of ATW that were issued (according to issue date) in Washington in 1864 and 1865.

