I'm not a native speaker of Japanese by any stretch of the imagination, but I gave this a shot:

”今夜、「真実二つ嘘一つ」（[2 Truths 1 Lie]）というゲームをやってみたいと思います。

皆さん、一人一人、自分について三つのことを紹介してください。その中の一つは嘘を付いて下さい。そして周りの人はその３つのことの中のどれかが嘘だとを当ててみてくださいね。当てるのを手伝うためにどんな質問をしてもかまいません。”

Let me know if you've got any questions and I hope a native speaker also comments. ^^ Good luck with the activity.

A man arrived at home. 

His wife asked him: Hey, did you buy food for the unicorn?

He answered: What are you talking about?

She angrily replied: You forgot to bring food for the unicorn, it hasn't eaten anything since this morning.

He said to her: Have you been smoking something?

With that, his wife suddenly began to destroy everything that crossed her path; some neighbours called the police based on the noises they heard. When the police arrived, they could see the woman angrily destroying everything and shouting "where is the unicorn's food?" The police promptly dragged her to the madhouse.

There, the psychiatrist made his analysis and found that she had mixed the real world with the imaginary world; reading lots of fairy tales seemed to be the cause. Ever since then she has had to wear a strait jacket to prevent her violent behaviour.

Business discussion course at an English school in Japan

I have been taking a business discussion course at an English school in Tokyo (this begs a time reference since you have present perfect continuous - so, something like 'lately' would work well). Usually, a recent news article is distributed to the students a week before the class and about 3 to 5 students then discuss in the class for 50 minutes based on the questions attached to the article.

I always wonder how I should behave in the class. I want to state my opinions as much as possible for practice, but at the same time, I want to listen to the other students' opinions with respect and share the time evenly. In the latest class ('This most recent class' works too), we had an interesting discussion but maybe I spoke only 5 minutes or so, which made me feel a little bit uncomfortable. I knew I could speak more but didn't want to be too selfish. Since listening to others carefully without interruption is thought to be polite (here/in Japan), if I kept talking, everyone would listen to me patiently...

Now I believe ('Through this I've learned' is maybe better) that the best way to practice speaking as much as possible must be is by taking private sessions. Until the discussion course ends by in October, I will focus on making the class more interactive by asking questions and giving responses so that everyone, including me, can have more chances to speak out. (The way this last sentence is phrased makes it sound like you're the professor of the class; I'm assuming you're a student taking the class instead?)

ひとまず「國際交流イングリッシュキャンプ」ではなく「国際交流イングリッシュキャンプ」です。読み方は「こくさい・こうりゅう・いんぐりっしゅ・きゃんぷ」です。宮城県（みやぎけん）における言語合宿みたいな場所みたい。

The first one should read 国際 rather than 國際, and is a summer camp held in Miyagi-prefecture to help kids learn English (it lit. means International Exchange English Camp).

で、「宮城復興支援センター」は「みやぎ・ふっこう・しえん・せんたー」です。宮城県（みやぎけん）を回復するための援助場所。 

This one is a public arm of the government in Miyagi-prefecture that attempts to raise funds in order to help recover lost or failing aspects of the local culture (as deemed necessary by the local government there).

Here's a link to their respective websites:

http://www.ganbaro-miyagi.com/

http://www.ganbaro-miyagi.com/esc/