How is Kurisu alive in the Alpha line?
This is a complicated question, but it basically comes down to Suzuha.
By sending the first D-mail, Okabe accidentally tips off CERN about time travel. Armed with that information, SERN develops their own time machine and eventually takes over the world (I'll get back to this point later).
Operating on instructions from her father, Suzuha takes a makeshift (and, because of SERN's monopoly on time travel, low-quality) time machine back to 2010 in an effort to stop SERN's rise to power. Because her co-ordinates are slightly off, she materializes in the roof of the Radio Building, rather than on it. Because of this, Dr. Nakabachi's conference is cancelled, he and Kurisu never meet there, and she doesn't get stabbed.
How did CERN develop time travel?
One of the steins;gate mangas, Shijou Saikyou no Slight Fever Chapter 3 "Muffled Phone", reveals that Kurisu went back to the Radio Building a few days after joining the Future Gadget Lab, and leaves her time travel thesis (the one she wanted to co-publish with Dr. Nakabachi in the Beta Line) there, where it is stolen by Moeka:
Presumably, Okabe's D-mail puts SERN on Kurisu's trail, and CERN uses this thesis to develop their time machine.
It's also strongly implied (if not stated outright) that the members of the Future Gadget Lab are taken into custody by SERN after they complete the Time Leap machine; Suzuha's intervention gives Okabe an opportunity to use the Time Leap machine and change his fate, but without her then SERN would have had more than enough resources to create their Time Machine, take over the world, and create the Bad Future that prompts Suzuha to go back and disrupt everything.
I hate temporal mechanics.
Okay, so what does deleting that D-mail do?
This is an odd case, and not explained very well.
Not having this D-mail in their database somehow prevents SERN from successfully developing a time machine1.
If the "jellyman" files are any indication, SERN's own time travel research was a stunning failure, and they would almost certainly have abandoned it. Kurisu's thesis is likely very theoretical, and wouldn't have given them the crucial insight that allows the Future Gadget Lab to develop the Phonewave and Time Leap: the notion of sending back data rather than matter.
It seems as though SERN really needs to get their hands on those devices, as well as the three who invented them (Okabe, Kurisu, and Daru) in order to successfully develop time travel. Without it, we end up in a different future where Suzuha:
- Still needs to travel back to 2010
- Has a better time machine (because SERN doesn't have a monopoly on it anymore) and is able to land on the Radio Building without crashing
With Suzuha not crashing, Kurisu is able to give Nakabachi her time travel thesis, which ends up getting her killed and getting us back onto the Beta World Line.
1 It's the how of this that I can't rationalize. I have a couple of theories, but none of them really pan out:
CERN hadn't yet looked at the text when Daru deletes it. Within the show itself this is plausible; Moeka appears to stumble onto the Future Gadget Lab entirely by accident, and then sticks around once she realizes what they're doing. However, the manga referenced above makes this a hard theory to swallow; it really seems like Moeka was specifically targeting Kurisu, which she'd have no reason to be doing unless ordered to by CERN. This theory is supported by an episode Q&A on ibm5100.net. However, this appears to be a fansite, and is about as canon as anything I've speculated on.
On reflection, it occurred to me that it's entirely plausible for Moeka to have been assigned to tail Kurisu and/or Dr. Nakabachi independently of the D-Mail, especially if CERN somehow became aware of Kurisu's time travel research. As far as I know there's no evidence for this, but it does somewhat redeem the theory.
CERN follows up on the lead, but ultimately decides that it's a dead end. Obviously the Future Gadget Lab isn't going to be pursuing any more time travel experiments after Okabe went through so much trouble to undo all those D-mails, so it's possible CERN eventually told Moeka to stand down after nothing came of this development. However, if that's the case then why bother deleting the D-mail? Okabe could just swear off time travel research, and we'd end up with the same result.
Unfortunately I can't come up with a satisfying explanation. "Because story" is, I suppose, the only thing left that makes sense.