I haven't seen it myself, but I hear it's really good. Pixar films are usually fantastic.

Here are my corrections...

I watched the movie, "Inside Out"

I watched the movie, Inside Out with my family.

Consider putting the two sentences together as:

I watched the movie "Inside Out" with my family. Much simpler!
インサイドアウト(邦題)を家族と見ました。

  It was SO exciting and interesting.

(This is more grammatically correct)
すごくおもしろかったです。

I'll get the DVD, and I'll be sure to watch it in English, without Japanese subtitles!

"I'll never fail to watch it in English" sounds kind of funny, honestly. :D But it should "I'll be sure to". And credits is wrong - should be subtitles!
DVDを買ったら、絶対日本語字幕なしで英語でみるんだ！

Twitter news says that green pepper in Japan is broccoli in the US.
Twitter でみたんだけどピーマンはアメリカではブロッコリーなんだって。

 Does this mean that many kids in the US hate broccoli?

Also, yes, it's true. :P
つまりアメリカではブロッコリーが子供達に嫌われてるということかな。

Disgust in the US is translated to ムカムカ in Japan.

Hmm, I wasn't aware of that. ムカムカ is like "feeling sick", right?
Disgust は日本ではムカムカと訳されています。

The Japanese word ムカムカ means not only ムカムカ(disgusted) but also イライラ(irritated).

This is similar to English, too. If we are irritated with someone, we can say they "disgust us".
日本語のムカムカには、おえ〜とか最悪〜とかいう意味だけでなくイライラするという意味もあります。

Thanks for include the original Japanese, too. It's useful for my own Japanese studies, and helps clarify your meaning.

Hope that helps!