I'll try and help you with what I can.
Like I said before in the F4 thread, the Ultimate titles are all a different universe than the mainstream Marvel Universe. In the Ultimate line, they restarted everything fresh, a new modern take on all your favorite characters. When you first see a character introduced in an Ultimate comic, you are seeing them for the first time ever. As far as telling which comics are in the Ultimate Universe, if it is, it will have the word Ultimate in the title. So Astonishing X-Men for example is in the normal Universe.
Ultimate X-Men - Vaughan and Immonen - This book is always cool to read, although there have been lapses in the artwork. Latest story was the introduction of Longshot and Mojo. The teammates would consist of Wolverine, Cyclops, Marvel Girl, Iceman, Kitty, Colossus, Nightcrawler, Dazzler, Storm. In the Ultimate Universe, Rogue has run away with Gambit (who has never been on the X-Men). And Beast is dead!!
In the normal Marvel Universe, you've got:
X-Men and Uncanny X-Men
X-Men - Peter Milligan and Sal Larocca - Awesome book, really talented artwork. A normal lineup would consist of Havok (leader), Iceman, Polaris, Wolverine, Gambit, Rogue, and Emma Frost. The latest storyline (Milligan's first) was Golgotha, giant mushroom slugs driving people bonkers. This has really brought the Gambit-Rogue relationship to the forefront, and the Havok-Polaris-Iceman love triangle to bear, you see Lorna Dane is now Iceman's woman.
Uncanny X-Men - Chris Claremont and Alan Davis - Okay book, Pretty good artwork. Never been as high on Davis' work or Claremont's storytelling as others. A normal team would be Storm, Bishop, Nightcrawler, X-23, Psylocke, Marvel Girl, Wolverine, Cyclops, Emma Frost. The latest storyline was the attempted takeover of the planet by a Savageland race evolved from Dinosaurs called the Hauk'ka.
General Story Trends: Professor X no longer lives in the mansion, he is presiding over Genosha in the capacity that Magneto had since Mags is supposedly dead again.... So Emma Frost, the White Queen fulfills the Prof's role. She is the head mistress of the school and the person who helms Cerebra (a more State of the Art version of Cerebro). Cyclops has taken responsibility of personnel decisions and is the highest in the chain of command, he doesn't really go out on missions much. And by the way, his sweetie is the White Queen, Jean Grey (Phoenix) sacrificed her life in Uncanny #450. So the Marvel girl you see on the Uncanny team is actually her daughter from a different reality, Rachel Summers. Due to the popularity of Wolvie, he's in both titles pretty regularly. Psylocke thought to be deceased in X-treme X-men (issue 2) has recently been resurrected in the Hauk'ka storyline. Colossus has been dead for quite some time, now....Uncanny X-men (issue 390). His sacrifice ended the threat posed by the Legacy Virus.
Those are the three issues I've collected continuously and religiously, these others I don't collect so I'll have less info.
Astonishing X-Men - Joss Whedon and Cassaday - Don't collect this but I've heard great things about this book and some of you may know Whedon as the creator of Buffy the Vampire Slayer TV series (Sarah Michelle Gellar).
New X-Men - Forget who does this book but it is re-hashing the New Mutants idea. Remember that comic in the 80s that brought kids (Cannonball, Moonstar, etc.) and it was like the X-Men minor leagues, this is like the new version of that.
Now about your question as to whether any of the Wolverine continuity is still intact because they restarted the series, the answer is Yes. Around 2002, they decided to restart a bunch of the series from #1 on. They also came out with new ones, Cable became Soldier X. Bishop became District X. Now while the numbers were reset, none of the backstory has changed on these characters so I doubt they erased all of Wolverine's past. And No, he doesn't have bone claws anymore, somebody kidnapped him in his ongoing series and did the same thing to him again.
I would think the Marvel Knights Spiderman would be the same, but I'm not a 100%. Marvel Knights is the adult imprint of Marvel but still in the normal Marvel Universe (same continuity), sort of like Vertigo with DC.
The New Avengers title is the only Avengers title I'm aware of (in the normal Universe) and yes Spiderman and Wolverine are now on that team. Remember "The Ultimates" is the Avengers in the Ultimate universe.
House of M, haven't started reading it yet, but its one those stories that'll cross-over quite a few titles (Astonishing, New X-Men, Uncanny, Avengers).
That's all I got.