I'm excited, as the new game will be released next week.   OR   I'm excited for the new game to come out next week.

"To be on sale" generally means "available at a discounted price". If the game is becoming available for purchase for the first time, you can say "coming out", which is more casual, or "will be released", which sounds a bit more proper. I'll also add a case to David's point about when to use "the" vs. "a" here: if the game you're referring to is obvious from the conversation's context, you can still say "the". For example, if someone was talking about a particular game series, you could then say "I'm excited for the new game to come out", which obviously means you are talking about the new game in the series.

I stayed in Hiroshima for one night, in Kobe for two nights, and in Kyoto for three nights.   OR   I stayed in Hiroshima for one night, in Kobe for two nights, and in Kyoto for three nights.

Both are okay, but the second one sounds a bit more natural.

It makes me feel great when I do the right thing.   OR   Doing the right thing makes me feel great.   OR   I feel great when I do the right thing.

Again, all three are okay, but the last two are a bit more natural.

Hope this helps! Let me know if you have any questions :-)

Can you please check these sentences?

Can you please check these sentences?

I wrote in Japanese as well just in case.

I'm excited as a new game is going to be on sale next week．
You can use the definitive article here only if you clarify which specific game this is. For example: I'm excited, as the new game I wanted to buy is going to be on sale next week.

来週、新しいゲームが発売されるので楽しみです。

I stayed in Hiroshima for one night, in Kobe for two nights, and in Kyoto for three nights.

広島で一泊、神戸で二泊、京都で三泊しました。

I feel great when I do the right thing.
  There is no need for "makes me" in this form of sentence.
However, it could work in passive. Example: Doing the right thing makes me feel great!

正しいことをすると、とても気分がいいです。

Thank you in advance.