1.) How bugged are the quests from 1-80? (Specifically Icecrown and Storm Peaks) - (Are all achieves able to be gotten in Northrend?)
2.) How is the scripting for dungeons/raids? (At 80)
3.) Are classes broken?
1) I'd say over 95% of quests are functional (And I'm rounding down to the lowest multiple of 5). You tend to notice bugged quests and you get annoyed by it... But before that one you've done a hundred that worked fine.
I returned to WoW a few months ago, after taking a 5 years break from it, and there has been a massive improvement in the debugging of quests. What I remember from my high school years is that half of the quests that were not "Kill 12 fat boars" or "Collect 5 tiny Dragon scales" were bugged... But maybe I was playing on bad servers.
In the end, what matters the most is: You can hit 80 by doing quests and not grinding mobs aimlessly, on a x1 rate.
2) End-game dungeons/raids are fine. There are a few non-functional features here and here (e.g Ley-Guardian Eregos' Planar Anomalies dealing 250 damage per swing to the player instead of much higher damage to the drake) but similarly to the quests, a great majority is functional.
I think I've heard there's something wrong with Gluth regarding when he "eats" adds but I don't remember how it's intended to be, and I feel like the fight's difficulty is roughly the same as other bosses, so it's fine by me.
I've raided Naxx, Ulduar, Sartharion, Malygos, and so far I haven't noticed or heard anything wrong.
3) I assume by "broken classes" you mean "X class' Y spell/talent deals no damage or deals too much or doesn't apply Z debuff".
I haven't heard or noticed anything wrong regarding that matter but I haven't played all classes so I cannot guarantee everything works fine.
You can sort the bug reports by how recently they were posted (Click on "Updated") and look through it to see if people have been reporting anything regarding Spells or Talents. Alternatively, you may use the search function for class names, or spells and talents, and see if people have been reporting something regarding that class/spell/talent.