Although others have covered the basics, here's a few more tidbits.
First off, by the way, welcome aboard. You're going to love the new series. As you watched 3 through 7 you know full well that there will be incredible episodes, some below-average episodes, and a lot of just-plain-good. Some reviewers and fans tend to amplify one over the other, but it's really business as usual for Doctor Who!
I don't know why I had low expectations
Given the generally shoddy track record of TV remakes, that's not an unreasonable initial impression of any remake. There were low expectations of Battlestar Galactica, too. The big different with Doctor Who is it's not a reimagining, or a reboot, or a remake. It's a direct continuation and a reinvigoration. Big difference over, say, Bionic Woman.
- I see that the DVDs are broken down into season like most TV shows but I also understand that there are several TV movies/specials as well that might not fit into the actual seasons. Is there a good online reference for the viewing order of the these specials (I assume they fall between seasons but I really don't know)?
The general point has been answered but you need to also know that the new era also introduced the "mini-episode". These were short, 7-minute stories that were produced for charity or special events. And at least 2 of them are considered canon.
After the final episode of Season 1 and the first episode (The Christmas Invasion) of Season 2, you need to see "Children in Need Special" (so-titled because it was part of the Children in Need charity appeal of 2005). The good news is it's on the Season 2 DVD set.
Between the season finale of Season 3 and the Christmas special that followed (Voyage of the Damned), there is a mini-episode called "Time Crash". This episode, by Steven Moffat, is a must-see if you're a fan of the Fifth Doctor, for reasons that will become immediately apparent! It's in the Season 4 box set.
Lastly, there was another mini-episode called Music of the Spheres. This one is hidden away as part of "Doctor Who at the Proms", a concert special included in the DVD release for the 2008 Christmas special, The Next Doctor, and also included in the 2009 Specials box set. It may or may not be canon, and can be viewed anytime, really, during the Tennant era. But it aired after the Season 4 finale and before The Next Doctor special.
Also in the category of mini-episode, for Season 2 the BBC produced "TARDISodes", short 1-2 minute webisodes helping to set up each week's story. Most of them take the form of prologues. Unfortunately, they weren't included on the DVD so you need to check YouTube or elsewhere for those when you get up there. They are not obligatory viewing.
The spin-offs haven't been addressed. There are 3 spinoff series. Torchwood has had 3 seasons. Season 1 needs to be watched at some point after the end of Season 1 and before the final three episodes of Season 3. Season 2 of Torchwood takes place after the Season 3 finale of Doctor Who. Season 3 of Torchwood also known as the Children of Earth mini-series should be seen after Season 4 of Doctor Who and preferably before the specials. It's a very different show from Doctor Who, and the first season is treated by Who fandom in much the same way Trekkies treated Enterprise, so your mileage may vary.
As for The Sarah Jane Adventures, Season 1 needs to be seen after Season 2 of Doctor Who, while Season 2 of Sarah Jane can be watched pretty much any time after that. Season 3 isn't on DVD yet but technically should be viewed before the final Doctor Who special, but that won't work because the DVD isn't coming out till the fall. Sarah Jane is aimed at a slightly younger audience than DW proper, however I found the storytelling in the first two seasons to be a lot more mature than I expected and several stories have ranked higher than some Doctor Who episodes in my opinion.
The third spin-off is a non-BBC production called K-9 which isn't on DVD yet. It takes place after The Invasion of Time (yes, that far back), so it can be viewed whenever. Unlike the other shows there's been no crossovers as yet.
On top of all this there were also two animated episodes produced. The Infinite Quest takes place sometime during Season 3, while Dreamland fits into the interval between the final 2 specials. Both are on DVD in the UK, while only Infinite is on DVD in North America, with Dreamland coming in the fall.
Confused yet?
Are there any episodes or specials that are "skippable"? Maybe the ones that weren't so good? Or are they all reasonably enjoyable?
You'll get different answers from different people on that one. The fact is each season has had an overall arc of some sort, so missing one episode means you miss a bit of the arc. In seasons 2, 3 and 4 the show instituted what were called "Doctor lite" episodes in order to allow the show to keep to its production schedule. They are what they say on the tin - the Doctor appears in only a few scenes of each. The first of these, Love and Monsters, is fairly reviled by fans. But the second, Blink, is possibly the most acclaimed Doctor Who episode ever and is probably the reason why its writer, Steven Moffat, now runs the show. And the third, Turn Left, was a vital part of that season's story arc.
I will say the animated episodes I mentioned, and the Time Crash and Music of the Spheres mini-episodes are optional. You can also get away without watching Torchwood or Sarah Jane though a few references in the main series might go over your head. And K-9 is meant to be a standalone, so you can choose to watch it or not. As I mentioned those TARDISodes are clearly optional as the BBC didn't even bother putting them on DVD.
But with only 13 episodes per season (14 including the Christmas specials), there's really no need to skip any of them.
I'm about halfway through Season 1 (just finished the one where they prevent Rose's father's death and then the WWII two parter. Both stories were fantastic). I understand that there have been two additional Doctors since the "current" one (from my perspective). If I recall correctly wasn't there a "12 regeneration limit"? Is this still considered canon or did they get rid of that? And if so it looks like they are heading toward that limit at a fairly quick pace.
Even back in the 70s and 80s everyone assumed they'd find a workaround. After all, the Master managed it. They could also decide to retcon it away, though they've admirably avoided doing too much of that (you may have noticed they've fiddled around with the math regarding the Doctor's age).
Without giving away any spoilers, does the "Time War/No More Time Lords" backstory get fleshed out or does it just kind of stay in the background?
All of the above.
And is Doctor #8 from the American TV movie still considered canon? (Hopefully without the half-human BS.)
After some initial doubts, an on-screen reference in a later season made the connection official. (There have since been a total of three direct references made). A piece of technology introduced in an upcoming (for you) season was used by the Doctor Who comic book to suggest a reason for the half-human thing. Russell T Davies was going to address it directly at the end of Season 4, something about "I was half-human for about a day", but cut it from the script before filming because he felt it would confuse new viewers. (This comes from his book The Writer's Tale: The Final Chapter which I STRONGLY recommend reading when you get caught up.)
And do we ever get details on the regeneration from #8 to #9?
No, and it's doubtful we ever will. At least on TV. At one point the Doctor Who Magazine comic strip was given the authorization by RTD to depict the regeneration, but the plan fell through. Without spoilers, as the series goes on you'll be able to piece together a likely scenario. Indeed, if you're already past "The Doctor Dances" you've already been given major clues.
The idea of Paul McGann appearing on Doctor Who as the Eighth Doctor is a rumor that appears like clockwork about every six months, especially if a "special episode" is announced. Although McGann continues to play the Doctor for BBC Radio/Big Finish's audio dramas, it's increasingly unlikely they'll bother having him appear on the main show.
Anyway, as I said, I'm enjoying season 1 so far. I'm glad I decided to give the series a chance.
And you ain't seen nothin' yet. It's a fantastic season, but the show really finds its legs in its second year. (And when you get to the fourth season, the exclusive-to-audio adventures kick in. Oh, and did I mention the full-cast Torchwood radio plays, and the Season 5-tie-in computer games ... I've gone cross eyed!)
Alex