Warning: Fanboy plot reinforcement ahead. Don't take it too seriously - I don't. My point is merely that there can be explanations, not that these are necessarily "correct" ones (it's fiction, after all).
If Rose was able to destroy the Daleks by simply having a long look at the TARDIS core, then why did the entire Gallifreyian species die out in the Dalek conflict? Were none of them able to do the same?
This is indeed one of the less strongly reinforced plot points. There may be more explanation to come - the writers don't always tie up every loose end immediately. Continuity works both ways - sometimes you leave hooks for future use or explanation. Perhaps the Doctor's old-model TARDIS, which has been in (active) service far longer than any other known one, has grown stronger?
Why was the Doctor able to survive the exposure?
He wasn't. He had to regenerate. OK, so it's survival of a sort, but not to be taken lightly. Some past plot points suggest the regeneration process isn't entirely stable, and has elements of mortal risk, sort of like chemotherapy today. You'll heal or die, and you can never be quite sure which.
If the captured Dalek destroyed itself because of the contamination from Rose, then why did the God Dalek consent to use humans as raw material?
Rose's DNA wasn't "sifted and filtered" like the processed Human-Dalek material. You may also notice that the mutant Kaled creature inside the Dalek engine destroyed in the TARDIS looked noticeably different from the two original-species Daleks we saw in the series (the captured one and the self-appointed god).
Why did Rose choose the words "Bad Wolf?" Why were they significant? Was there any useful meaning?
There was no meaning in itself. But it's a paradox. It was the words she saw (which just happened to be "Bad Wolf") which made her realise there was still a connection, a possibility, that she was still involved in the Doctor's world. When she made her desparate attempt and gained the powers to do what she did, she realised that the thing that had led her here was those words. So she took the words - the same ones which she knew worked - and scattered them where she'd find them, thus completing the circle. If she'd scattered different words - even meaningful ones - the timeline might have played out differently, and that would have led to a paradox which didn't come full circle, leaving loose ends, which would be inviting the reapers in.
Why would the Autons, the Rift, the Slovenes, the gas creatures, and Rose's home all be in Cardiff? I'd never heard of this place before; the coincidence strains credulity.
They weren't.
Rose's flat and the auton invasion were both in London. Some of it was filmed in Cardiff, some of it wasn't, but it was set in London. You might as well ask why all Trek adventures take place in a studio on Paramount's lot. Within the context of the story, they don't.
The Gelth (gas creatures) were at the same location as the rift because they found it there and were using it to get to Earth. The rift was in Cardiff by pure chance - it had to be somewhere, and as it happens, it was there. This is even a plot point... the rift caused strange things which freaked people out, and were this reality it might be precisely why you hadn't heard of Cardiff... it's a place where Weird Shit happens and people don't want to have much to do with it, despite business prosperity there. Think Sunnydale, in Buffy - why wasn't that place broadcast to the world?
The Slitheen were originally in London. The one surviving Slitheen needed to move far away from London after the destruction of 10 Downing Street, but it would make sense to stay within a culture she'd come to know and gain influence in, as it would be quicker to rebuild than to start again in another culture. Wales was one option, and also had the rift, which might present other opportunities for escape from Earth.
More attention seems to have been paid to continuity in the old series. It needs better plots.
It's not perfect. But it's not aimed at those who pick holes in everything. There are other SF dramas that are. But Doctor Who is aimed at being fantasy/SF fun for families. And in that goal, it succeeds admirably.
It will be interesting to see what Torchwood does, since the target audience is quite different. Anyone who wants to know just how far BBC3 will go should take a look at "Bodies" - a medical drama which is set in a gynaecological ward, and covers the machiavellian politics and medical issues that happen there in the most graphic detail I've ever seen on any TV program (including documentaries which show the real thing). Close-up detail (which is either real or completely convincing) of babies emerging from their mothers while doctors frantically cut the mother's genitals open to allow the baby to pass a constriction... it's quite shocking, and not to be taken lightly. But none of this is gratuitous - it's simply not hidden or glossed over. This is what these people see at work every day, and this is why they're stressed out to near-snapping point. And seeing what they see adds a level of understanding that would be hard to achieve any other way.
BBC Three is gaining a reputation over here for incisive (no pun intended) drama and quality production. It's something of a niche audience - quality is definitely favoured over mass marketability - but this is sometimes a good thing. How often has a truly brilliant series failed because it didn't sell massive numbers, despite critical acclaim? Think of Firefly, for example. That's far less likely to happen here.
Torchwood sounds to me like it's in the right place.
Gosh, that was a longer post than I'd expected...
