Binding Washington? Ordered abbreviations

Question

Twelve lists of ordered abbreviations are given below. The first part of your mission is to find out what comes next in each of them. Further instructions will await you once you have completed this first task.

$$\begin{array}{align*} \textbf{ 1 }&\text{ $\implies$ }&\text{AL, AT, DR, FY, JY, JY, ?}\\ \textbf{ 2*}&\text{ $\implies$}&\text{SR, OR, NR, MY, MH, JE, ?}\\ \textbf{ 3 }&\text{$\implies$}&\text{FR, CH, US, DE, CH, NO, ?}\\ \textbf{ 4* }&\text{$\implies$}&\text{BR, GB, CN, GR, AU, US, ?}\\ \textbf{ 5 }&\text{$\implies$}&\text{NY, MD, GA, ME, TX, LA, ?}\\ \textbf{ 6 }&\text{$\implies$}&\text{PE, AB, NB, NS, NU, QC, ?}\\ \textbf{ 7 }&\text{$\implies$}&\text{JA, JQA, CAA, JB, GHWB, GWB, ?}\\ \textbf{ 8 }&\text{$\impliedby$}&\text{BO, JAG, GC, GF, FP, CC, ?}\\ \textbf{ 9 }&\text{$\impliedby$}&\text{AS, ZN, PB, AU, I, BR, ?}\\ \textbf{ 10* }&\text{$\impliedby$}&\text{HG, SB, CO, P, S, AG, ?}\\ \textbf{ 11 }&\text{$\implies$}&\text{MHALL, LL, LL, MHALL, WFWWAS, AETMW, ?}\\ \textbf{ 12 }&\text{$\implies$}&\text{AYS, AYS, BJ, BJ, MBAR, MBAR, ?}\\ \end{array}$$

Fill out your answers in the grid below and please note that they are not in numerical order.

                                                    

Once you've found the answers and filled the grid, follow the steps below to complete your mission.

Step 1:

In the grid, after which numbers might the characters ':' (I) and '&' (II) follow?
Combine these numbers -(I) and (II)- to form (III) and divide the second key by (III).
Now, take the square root of the number you've just obtained and call the result $n$.
Count off every $n$th character of the message below. When you reach the end, start again from the beginning, until you again have a 33 character message (the question mark does not count and stays at the end).

Step 2:

Reverse the first key and transform (5,4) to obtain a final nine-letter key.
Use the resulting key to decrypt (Vigenère) the rearranged message from Step 1.
Add spaces and punctuation to find the final question.
Finding the answer to this question will complete your mission.

Message:

UDTCMIOLRQIFCIAITAHSZARORHREKVAOC?

Answer 1

I think the asterisk denotes ...

... decreasing series. But what's the arrow for then?

1:

JE for June. The months are sorted alphabetically and the abbreviations are the first and last letter. (Thanks to Techidiot for helping me see the rather obvious sorting criterion.)

2:

JY for July. Same as a, but sorted backwards.

3:

IT for Italy. These are the host countries for Olympic Winter Games.

4:

ES for Spain. These are the hosts of the last Olympic Summer Games, but backwards.

5:

ND for North Dakote. These are the US states sorted alphabetically by their capitals.

6:

SK for Saskatchewan. The provinces and territories of Canada sorted by capital.

7:

JC for Jimmy Carter. These are the US presidents alphabetically by family name.

8:

JM for James Monroe. The pattern is the US presidents sorted by the family names reversed (amabO, dleifraG, dnalevelC, ...). This doesn't have an asterisk, because the sorting order is forwards, only the criterion is backwards.

9:

CL for Chlorine. The symbols of the chemical elements sorted by their names backwards (cinesrA, cniZ, daeL, dloG, ...)

10:

CU for Copper. Sorted like 9, but the last elements from the bottom up.

11:

MW for "Mary went", the next words in the nursery rhyme Mary had a little lamb. This one was found by Rubio.

12:

DDD for "Ding, dang, dong!", the next line in the nursery rhyme Brother John, the English version of Frère Jacques.

It turns out that an arrow pointing left ...

... denoptes series where the names to sort are backwards. (The OP pointed me towards this fact in a comment.)

The completed grid looks like:

         6       S K
         1       J E
         2       J Y
         4       E S
         5       N D
        10         C U
         9         C L
         8       J M
        11         M W
         7       J C
        12       D D D
         3         I T

Giving:
  KEYS: DCC & MMCDI

where the keys are Roman numbers. The punctuation is inserted after 4 and 9, which combines to 49. (Note how the question says "In the grid, after which numbers might the characters ':' (I) and '&' (II) follow? Combine these numbers ...")

MMCDI is 2401, which, when divided by 49, gives 49 again. The square root of 49 is 7. That makes the cipher text for the Vigenère:

    OIZEDRIRVCIAROICSRULAAKTQTOAMFHHC

The key for the decryption is:

JAMES BOND. The first Roman number is DCC or 700, which is 007 reversed. The enumeration of letters (5, 4) fits. Decoding yields:

  Final question: Who follows SC GL RM TD PB?

The answer to this is:

DC for Daniel Craig. These are actors who played James Bond: Sean Connery, George Lazenby, Roger Moore, Timothy Dalton and Pierce Brosnan.

And, of course, DC is the District of Columbia, which explains the title.