Мне было 6 лет
This is an example of the so-called possessive dative (дательный притяжательный). It is used to describe relationships он мне друг "he's my friend", literally "he's a friend to me", она мне мать "she's my mother" etc.
Idiomatically, it's used to describe age as well. Romance languages, like Italian, also use possessive constructs to describe age: ho 6 anni "I'm six years old" (literally, "I have six years").
Мне было холодно
This is an impersonal predicative construct (безличный предикатив). They're used to describe feelings, emotions, states etc.
If you think about it, the English phrase "he's cold" can mean two things: it can mean that he literally is cold (has low body temperature), or it can mean "he's feeling (the sensation of) cold". In English, you can only distinguish these meanings from the context; in Russian, they use a different syntax: он холодный (his body is cold) vs. ему холодно (he's feeling cold).
Other predicatives like this are ему страшно "he's afraid", ему грустно "he's sad", amongst others. You'll have to learn them by heart.
Он был хорошим человеком
Она была веселой
These are personal statements with composite nominal predicates (личные предложения с составными именными сказуемыми). A composite nominal predicate consists of a verb and a nominal (noun or adjective) the verb governs.
Verbs like быть "to be", стать "to become", являться "to be" (literally, "to present oneself") govern the instrumental case.
The verb быть can also govern the nominative, and it's one of the rare cases where the feature of definiteness surfaces on a syntactic level in Russian: весна будет тёплая means "it will be a warm spring"; весна будет тёплой means "the spring will be warm". Стать and являться can only govern the instrumental.
Всё было хорошо
This means "all was well", just like in English. Хорошо here is an adverb, not a short adjective.
