13. Vyxal, 20 bytes

91726128185f7g22V¦C∑

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Prints \t\n "()+34<A. Next answer should print \t\n "()+124<A.

The deltas between the character codes are [9, 1, 22, 2, 6, 1, 2, 8, 1, 8, 5], which conveniently doesn't include either of the two banned digits - so we can just take the digits of 91726128185, replace the 7with a 22, and take the cumulative sum before casting into characters.

12. Ruby, 53 bytes

$><<[?\s<<65<<3*3<<43<<5*2<<40<<43-2<<34<<7*7<<57][0]

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Prints <space>A<tab><newline>()"19. Next set: <space>A+<<tab><newline>4()"3

11. ACCUMULATOR, 54 bytes

Using bb94's answer: <newline><tab>19<space>"()+\. Next set: <space>A+<tab><newline>()"19

ACMUMUAAAAAAAAAAORAAACMUAAUAAAAAAUAUAAURAAAAAAAACAAAAU

10: Raku (Perl 6) (rakudo), 74 bytes

().^lookup("{--($="d")}h{--($="s")}s")((9;32;43;44;49;92;99;89-(-25))).say

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Prints <tab><space>+,1\cr<newline> (the newline is a trailing newline). Next set: <space>+<tab><newline>()"19\

8: Python 2, 34 bytes

exec"p\162int'\40\t\53\50\51\162'"

Explanation:

exec                               # run (in Python 2, this is a statement)
   "p\162int'\40\t\53\50\51\162'"  # <space><tab>+()r (r is replaced twice)
                                   # and then a newline (which is implicit)

Prints <space><tab>+()r<newline>. Next set: <space>+<tab><newline>\rce

9: Brachylog, 43 bytes (37 characters)

[32,43,9,92,114,99,101,10]~ạw

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Prints <space>+<tab>\rce<newline>. Next set: <space>+<tab><newline>1\r,c

7: Whitespace, 73 bytes

^{The byte-count is 1 below the limit of \$2\times7+60\$. :)}

[S S S T    T   T   S S T   N
_Push_57_c][S S S T T   S T T   T   N
_Push_55_a][S S T   T   N
_Push_-1_)][S S T   T   S N
_Push_-2_(][S S S T S S T   S S S N
_Push_72_r][S S S T N
_Push_1_+][N
S S N
_Create_Label_LOOP][S S S T S T S T S N
_Push_42][T S S S _Add_top_two][T   N
S S _Print_as_char][N
S N
N
_Jump_to_Label_LOOP]

Letters S (space), T (tab), and N (new-line) added as highlighting only.
[..._some_action] added as explanation only.

Try it online (with raw spaces, tabs and new-lines only).

Explanation in pseudo-code:

Push the codepoint-integers minus 42 of string "ca)(r+"
Start LOOP:
  Add 42 to the top integer
  Print it as character to STDOUT
  Go to next iteration of LOOP

The constant 42 is generated by this Java program, based on this Whitespace tip of mine.

The next set is <space>+<tab><newline>r().

6: C# (Shell), 54 bytes

$"{(char)43}{(char)62}{(char)101}{(char)46}{(char)92}"

Prints +>e.\. Has to be in a shell rather than standalone, since there's no way to output in C# without e otherwise.

Next set is +r()ac (link)

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5: JavaScript (Node.js), 36 bytes

process.stdout.write("\53\55\74\76")

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This prints +-<>. I previously had a more interesting GolfScript solution here, but I realized that it contained a + which made it invalid.

Next charset is +>e.\ (link)

3: brainfuck, 42 bytes

++++++++++[->++++++++++>+<<]>+.[->+<]>+++.

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Outputs er. Can't fit in js

4: Uiua, 10 bytes

&pf-1$ ,=?

Uiuapad

New banned characters: +-<>

2: Keg, 2 bytes

Ɛ

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Simply prints the letter e and exits.

1: Python 3, 18 bytes

while True:
	break

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e - 3 (Now banned)
r - 2
<tab> - 1
<newline> - 1
<space> - 1
: - 1
T - 1
a - 1
b - 1
h - 1
i - 1
k - 1
l - 1
u - 1
w - 1

This code I stolen from ChatGPT makes easier to make the character list, but you have to combine answers manually.