What are the paintings in the Anti-Angel Headquarters and what do they mean? Here is a partial list available on Wikia. What do the ever-changing paintings mean and how are they connected to the episode's or the overall plot? For example:


   In the OP of Episode 4, hung on the side wall is the painting with the kanji [鎮魂歌]【ちんこんか】, which means requiem. The painting alludes to both the album Requiem for Innocence by Sad Machine that has a profound influence on Masami Iwasawa and her disappearance at the end of Episode 3.
Directions with respect to Yuri's perspective, where the front is facing the entrance. For the sake of completeness, I also include the paintings that are fixed across episodes.



All episodes



Front: School song of the Heaven Academy. The song lyrics reproduced below reference this yahoo answer and this wiki page.


  校歌
  
  緑豊かに広がる丘に
  光輝く我らが母校
  大いなる大地と
  天のかけ橋
  真実を知る時天に登らん
  
  心のままに戒めなくは
  かけ[●●●●●●●●●]【し希望にひるがえる】
  千里離れた空を思へば
  遠く忘れし思いはいづこ
  真実を知る時天に登らん
  
  光あふれる月の夜
  耳をすまして聞き入れば
  天使の歌声高らかに
  思いを胸に登る[●]【時】
  真実を知る時天に登らん
  
  天上学園


Back: Portraits of all past principals of the Heaven Academy. The portrait of the current principal has been replaced by Yuri's.

Left: During the operation meetings, the banner of the SSS Brigade is hanged on this side of the wall.

Right: Smiley face painting. Unknown artist.



Episode 1



Front: [危]【き】 — Danger.

Left: テトラポット — Tetrapod, a tetrahedral concrete structure used as armour unit on breakwaters.

Right: [南紀]【なんき】みかん — Tangerine from Nanki, Mie Prefecture.

Note: Jun Maeda, the original creator of Angel Beats!, is from Mie.



Episode 2



Front: [毒]【どく】 — Poison. I believe it's intended to invoke the expression 「お気の毒に」, meaning "my condolences", as said by Angel about Hinata in episode 5. By Jun's arrangement or by coincidence, Hinata was forced into sacrifice in an arithmetic progression of episode number 2, 5 and 8.

Left: [悦]【えつ】 — Rejoicing.

Right: [松阪牛]【まつさかうし】 — Matsusaka beef, another food product from Mie.



Episode 3



Front: [嫁]【よめ】 — Wife; bride; (one's) daughter-in-law. Don't know if this has anything to do with 俺の嫁 〜あなただけの花嫁〜 or not. 俺の嫁 is a love adventure game released in the same year as Angel Beats! and sharing some of the voice actors (Emiri Katou, Kana Asumi, Eri Kitamura and Kana Hanazawa).

Left: [諭吉]【ゆきち】 — Yukichi Fukuzawa, the guy on the 10,000 yen banknote. Yukichi is the nickname of Yuuki Hashimoto from the 2010 eroge 恋色空模様 who is also voiced by Harumi Sakurai (Yurippe's voice actor). Moreover, "Farewell, Yukichi" (「さらば諭吉」) is the defining quote of Kyousuke Natsume from Little Busters!, another visual novel written by Jun Maeda.

Right: マニア — Mania, i.e. Takeyama.



Episode 4



Right (OP): [鎮魂歌]【ちんこんか】 — Requiem. Its significance has already been explained in the question.

Front: [脇]【わき】 — Supporting role; the second stanza of a renga. Yui changed her role in the band from support to lead after Iwasawa left the Afterlife World. At this point in time, the story has also entered its second phase in the hokku→waki→daisan→ageku renga structure or the introduction→development→twist→conclusion Kishoutenketsu structure.

Left: (Not shown).

Right: [逆鱗]【げきりん】 —  A scale grown upside-down below the chin of a dragon that will anger the dragon when touched; imperial wrath. Yuri's underlings incurred her wrath by failing their mission of defeating Angel's team in the baseball tournament. There is also the angered dragon conjured up in Otonashi's mind in episode 7.



Episode 5



Front: [謎]【なぞ】 — Enigma; mystery, i.e. Ayato Naoi in this episode. If you turn up the volume just after Yuri throws her second tantrum, you can also hear a 'mysterious' voice saying "what's with her?" (「なんだこいつ」). This is documented under the 謎の声 section here.

Left: コテンパン — Black and blue (e.g. beaten ...), i.e. Angel and Hinata in this episode.

Right: バミューダ — Bermuda.



Episode 6



Front: [乙]【おつ】 — Second; the latter; strange. Describes Ayato Naoi well in this episode.

Left: (Not shown).

Right: [猫大陸]【ねこたいりく】 — Continent of cats. Presumably the same painting appears again in the second panel on page 4, chapter 30 of Angel Beats! The 4-koma: Our War Front March Song:



In this panel, Yusa is explaining the term "marriage hunting" (婚活) to Hinata. The true meaning of 猫大陸 still eludes me, but if I were to fashion a guess, it would at least be somewhat related to marriage hunting, which reminds me of the missed opportunity in this episode for Otonashi to do, I mean, to become closer to Kanade, and the fact that the 'marriage hunting' was initiated by Naoi on behalf of Otonashi, too!



Episode 7



Front: プラ — Recycling symbol used in Japan for plastic.

Left: [濃厚]【のうこう】 — Rich (in flavor, color, smell, etc.) and strong; very likely; passionate and hot.

Right: ●(●)ック? (Not legible).



Episode 8

Front: (Not shown).

Left: (Not shown).

Right: (Not shown).



Episode 9



Front: ♨ — Icon for hot springs.

Left: [部族]【ぶぞく】 — Tribe; clan.

Right: マグロ? (Not fully shown) — Tuna.



Episode 10



Front: (Not shown).

Left: ●ロス? (Not legible).

Right: [赤福]【あかふく】 — [赤福餅]【あかふくもち】, wagashi of Akafuku shop, Ise City, Mie Prefecture.



Episode 11



Front: (Not shown).

Left: (Not fully shown).

Right: [極上]【ごくじょう】 — First-rate; best.



Episode 12



Front: [心]【こころ】 — Heart. Lots and lots of hearts on the computer screens.

Left: (Not shown).

Right: [土]【ど】● (Not fully shown).



Episode 13



Front: [魂友]【こんゆう】 — Spiritual friend.

Left: (Not shown).

Right: (Not shown).



Special: Another Epilogue



Front: (Not shown).

Left: (Not legible).

Right: [東]【とう】●? (Not legible). It would make more sense if the character is [転]【てん】, though, as that means "altering pronunciation or meaning". 



OVA 1: Stairway to Heaven



Front: (Happy face).

Left: (Teeth-gritting face).

Right: (Happy face).



OVA 2: Hell's Kitchen



Front: [再]【さい】 — Again.

Left: ボンレス — Boneless.

Right: [久々]【ひさびさ】 — (in a) long time; long time (ago). It's been almost 5 years since the last Angel Beats! episode was released.