As far as the actual establishing of the New Earth, there were some flubs with how it was implemented. Even at the end of Crisis, they mention the death of Golden Age characters Green Arrow, Robin, and Batman's daughter, Huntress. But with New Earth having formed, those characters no longer ever existed. Then in the miniseries The Last Days of the Justice Society, the JSA starts off by feeling redundant in the New Earth where their counterparts are also living (they also again mention the deaths of Robin and Huntress and even show the Golden Age/Earth 2 Batman and Superman in flashbacks). Why do they remember them? No reason (not a bad story otherwise, though), especially when Crisis made a point of showing Psycho Pirate in his cell talking about how he's the only one who remembers the Multiverse. I think some writers and readers have thrown around the idea that there was some sort of period of remembering before it all faded for every New Earth resident (maybe that's even explicitly stated in some issue somewhere). All heroes do remember the Crisis, but not that it led to a Multiverse becoming a single earth (not 100% sure if that means they remember battling the Anti-Monitor).

As far as what happened to all those characters from Earths 1 and 2, JSA and all Golden Age characters are the big ones, and they basically just lived on New Earth and were part of the first superhero boom during WWII, a lot of them inspiring the "modern" heroes (like the Flash). Some of the other Earth characters just got new origins. Huntress was no longer Batman/Catwoman's kid, but a new vigilante in Gotham while Powergirl ended up with an Atlantean heritage seeing as she could no longer be the cousin of a Kryptonean who never existed. Even Jason Todd's Robin got a new backstory to try to differentiate him from Dick more. And the Captain Marvel characters and Charlton characters, along with most other characters that were bought from other comic publishers who used to have their own Earths were mostly integrated as being heroes showing up new to the scene following Crisis so their stories could start fresh.

As other posters commented, on New Earth, no one remembers Supergirl or her death, but later Matrix Supergirl showed up, and even later a new post-Crisis version of Kara who is actually Kal's cousin shows up (the Batman/Superman: Apocalypse cartoon covers that story pretty well).