What is the rule for using which pronouns when talking about emotions/feelings in Russian. For example, why is it "Я счастлив" but "Мне грустно"?
Are these just patterns that you have to learn off with every emotion? What about angry, disappointed, excited, sleepy, etc?
It depends on a part of speech that you use to describe your emotion. When we use an adverb (грустно), the construction is Мне + adverb (Мне грустно), but if you use an adjective, the construction is Я+ an adjective (Сегодня я грустный. or Сегодня я грустен. (it's a short form of an adjective).
Я +an adjective is similar to I am + an adjective (apart form adjectives with -ed ending) I'm sad/happy/angry etc.
Мне + an adverb is similar to I feel + an adjective with -ed ending (I'm bored, I'm excited.)
Each time you learn a new word to express your emotion you have to pay attention which construction to use according to its part of speech. For example, "angry" can be сердитый или злой so we say Сегодня я злой (зол - short form). Сегодня я сердитый (сердит). But if you are bored, you can't say Я скучный because it means I'm boring! you have to say Мне грустно.