よし、きめました！ = Good, it’s decided! (Or I’ve decided, depending on context)

よく　わらいました。 = (I) laughed often.

にほんご　の　べんきょう　を　はじめます。 I’ll start studying Japanese (べんきょう = study)

Yes, but you can also think of it as meaning “of”.

AのB = “A’s B” or “the B of A”. Basically the way that you use nouns to modify other nouns. When you’re using an adjective to modify a noun, you can just stick them straight together – あかい　ほん = a red book. When you’re using a noun to modify another noun, though, you need to join them with の. わたし　の　ほん = my book. Which book is it? A red book. My book. In both cases, the “book” is being modified by the words that come before.

にほんご　の　べんきょう = study of Japanese
にほんご　の　せんせい = a teacher of Japanese
にほんじん　の　せんせい = a teacher who is a Japanese person
かぜ　の　たに　の　ナウシカ = Nausicaä of the Valley of Wind
ながのけん　の　まつもとし = Matsumoto in Nagano Prefecture.

And so forth.