Charles, I think your writing is quite good.
Just some minor mistakes which I pointed out below.
Please note that the sentences I wrote is just as one of many possible way to express the same thing, they are certainly
not "the one and only correct answer" - everyone has his/her own taste and style. I'm sure other members would write differently.
二ヶ月前に英語先生の会議ではひどい授業を受けさせられました。授業は日本人の先生と気の合う関係にするの仕方についてべきだったけど、その代わりに、自分らの授業ではどうやって日本語のしゃれを言ったらいいか教えられました。発表の目的は英語の授業で日本語のしゃれを言うなら、生徒たちが前より先生にもっと注意をすることですが、ばかな考え方と思います。しかも、日本語のしゃれが分かる会議に来た英語の先生はいませんから、発表中（三時間！）みんなはそわそわ座って気まずくなってきました。
Two months ago at an English teacher's convention (I wrote meeting, but convention would be better), I was forced to attend an awful class. The class was supposed to teach us how to create good relationships with our Japanese coworkers (perhaps here I ought to have stressed that I am an ALT and used the term JTE), but instead of that, we were taught how to use Japanese puns during our class. The purpose behind the presentation was that if teachers learn how to tell Japanese puns during class, the students will pay more attention to them. But, I thought this was a pretty ridiculous idea. Furthermore, since there weren't any ALTs in the class who could understand the Japanese puns, we spent the whole meeting (three hours long!) growing restless in our chairs and feeling increasingly awkward. -= WOW, I think I tried to do too much in that paragraph. Sorry about that. Anyway, any help is appreciated! =-
You can say 英語教師 but with the word 先生 you need to use の.
気の合う means compatible, go along well, etc. and is used to describe about people, but not a good word to modify 関係.
"How to (verb)" can be translated ～の仕方 but it is noun that comes before の.
To use a verb, 連体形＋仕方。 いい関係を作る→いい関係の作り方.
Using verb instead of noun would make the sentence sound more natural, though.
べき is more like "should" "must" than "supposed to be".
I would use ～はず instead.
My attempt:
日本人の先生方と良好な関係を築くにはどうすべきかについてのはずだったけれど、
英語の授業で日本語のしゃれを言うなら、生徒たちが前より先生にもっと注意をすることですが、
注意をする means to warn. "To pay attention" in this context (students pay attention to the lesson) would be 集中する or ちゃんと聞く.
I would drop the word "purpuse"and "presentation" as it is obvious you're talking about the lesson, so I might say:
英語の授業で日本語の駄洒落を言えば、生徒たちはもっと先生の授業に集中するというのですが、
or
英語の授業で日本語の駄洒落を言えば、生徒たちはもっとちゃんと先生の話を聞くということですが、
(with というのです・ということです, I'm implying "according to them, it is... but I think it's a stupid idea.")
しかも、日本語のしゃれが分かる会議に来た英語の先生はいませんから、発表中（三時間！）みんなはそわそわ座って気まずくなってきました。
This sentence is a bit confusing, it is difficult to understand which part modifies which part.
しかも、会議に来た英語の先生で日本語の洒落がわかる人はいなかったので、三時間（！）もの会議の間、みんな（は）落ち着かず、次第に気まずい雰囲気になっていきました。
とにかく一つのしゃれしか覚えていません（一番分かりやすいのですから）。発表者は「アメリカの地理を教えたいすると。ネバダ州の東の州を刺しながら、何という州、これって言って。生徒が正解答えをしてから、生徒たちに向いて、「誰がいうた？」と言って。そうするとみんなはきっと笑うよ」と言いました。
Anyway, I only remember one pun (because it was the easiest one to understand). The presenter said: "Let's say you want to teach American geography. While pointing to the state next to Nevada, ask the students, 'What's the name of this state?' After a student guesses the right answer, turn towards the class and say 'Who said that?' (pun on Utah Iuta). If you do that everyone definitely will laugh.'
Right answer is 正解, 解 is equivalent to 答, and it is something one 言う, not する. (Or one "answers" i.e. 答える.)
発表者 is a correct word but somehow it sounds awkward. If I were you, I'd knock it off. Even without it, it is still understandable that you're quoting what you were taught at the horrible class.
とにかく、覚えた駄洒落はたったひとつでした。（一番分かりやすかったから。）
「例えばアメリカの地理を教えたいというとき、ネバダ州の隣りの州を指して生徒に「これは何州？」と質問する。生徒が正解（ユタ州）を言ったら、クラスに向かって「誰が言（ゆ）うた？」と言う。そうすればみんな間違いなく笑うから」
以上は彼の一番おかしいしゃれですけど、ゴミ同然と思います。とにかく、この文で文句を言うつもりではありませんでした。書いた訳は「言うた」について質問してもらいたいです。最近から友だちが言ったの代わりに言うたを使う気がつきました。でもそれはOKかどうか普通かどうか分かりません。教えてください！
Although the above was his funniest pun, I still think it was garbage. Anyway, my point in writing this was not to complain. I wrote this because I hope someone will answer a question about 'iuta.' Recently, I have noticed my friend say iuta instead of itta. But, I don't know whether or not that is acceptable or normal. Please help!
これは駄洒落の中でも一番ましだったものですが、それでもゴミ同然と思います。
ですが、文句を言うためにこの文章を書いているのではありません。書いた理由は「言うた」についての
質問に答えてもらいたいからです。最近友だちが「言った」の代わりに「言うた」を使う
ことに気がつきました。でもそれが普通なのかどうか分かりません。（or それがOKなのかどうか分かりません。）教えてください！
2) Is 言うた widely used as a slang equivalent for 言った? Or would I sound strange/wrong if I used it in casual conversation?
言うた is quite undrestandable and you can use it in casual conversation if you feel like to, but it would sounds strange. To me it sounds like a part of old man's speech. Something like;
あんたがそう言うたからここで待っとったんじゃが、いくらたっても誰も来んかった。
（あなたがそう言ったからここで待っていたんだが、いくらたっても誰も来なかった）
for example. ;-)
3) I was just talking with a couple teachers about the adidas slogan 'impossible is nothing' which someone had put on the cover of the undoukai brochure. One person was lamenting its use because of its incorrectness, and at the end of a long [discussion] I said, 'でも、相談の種ね、これは。' This very much confused them, and enventually it became clear that I should have said 論議の種 or 話し合いの種. What exactly was the issue with 相談？ Does it always carry the conotation that one person is doing something for another person? (ie one person is an expert on something and gives advice)?
As 種 means the cause or the material on which something is based on, you can say 頭痛の種、癪の種、心配の種、話の種、噂の種、etc.
To me 相談 (something between consultation and discussion?) does not fit in.