For a lot of English speakers, the "had" and "have" in contractions is completely omitted in certain contexts. It's more prevalent in some dialects (I'm in the south US and it's more common than not). Usually "had" is dropped more than "have".

Also, English can drop the pronoun, article, and even copula for certain indicative statements. I think it's specifically for observations, especially when the context is clear.

looking at someone's bracelet "Cool bracelet." [That's a]

wakes up "sigh Gotta get up and go to work..." [I've]

"Ain't no day for picking tomatoes like a Saturday." [There]

"No war but class war!" [There's]

"Forecast came in on the radio. Says there's gonna be a hell of a lot of rain today." [It said -> Says/Said]

"Can't count the number of Brits I've killed. Guess I'm just allergic to beans on toast." [I; I]

"House came tumblin' down after the sinkhole opened up" [The]

"I'd" can be "I would", mainly if used with a conditional or certain conjunctions/contrastive statements (if, but, however, unfortunately). Also when preceding "have" – e.g. "I'd have done that". Because "I had have" doesn't make sense, nor does "I had " anything. "I'd" as in "I had" is followed by a past participle.

"I'd" is usually "I had" otherwise, forming the past perfect tense. But in "I'd better", it's a bit confusing because "had better" is used in a different sense – the "had" here comes from "have to" (as in "to be necessary to") and can be treated as both a lexical verb and an auxiliary verb. "had better" is a bit of a leftover of more archaic constructions.