1. (Nana is 5 frames behind in Brawl, unlike Melee's 6) Predict and shield 5 frames early. Popo will regular-shield, while Nana will powershield. There's not a whole lot you can do about that, Ice Climbers are naturally at a 5 frame disadvantage when it comes to any kind of reaction.
2. There are a number of scenarios where you need and needn't go for Nana. The first is any kind of hit where both Climbers can't recover without Belay/Squall (you'll learn this distance in time). Obviously in these cases you should always go straight for Nana unless your opponent is specifically trying to stop you from saving her.
The second is where you don't need Belay/Squall, but your opponent breaks away in order to directly kill Nana. Unfortunately there isn't a whole lot you can do here, especially if they are using a character with a great air game (MK, Kirby, etc.). In these cases you should try to pressure your opponent and hope to knock them away or at least distract them enough so that he doesn't KO Nana. Don't follow them out over the edge in any instance, no matter how easy you think it will be to get back onstage afterward. You will die.
Finally, there is when Nana can recover by herself and isn't being targeted. In these cases you should let her recover by herself and play super-safe in the meantime, but also try to put pressure on him so he actually has to deal with you and can't just cop-out and go for Nana. When she's on the ledge, go synch with her ASAP cause she tends to do stupid **** when left alone.
3. lol. Don't approach with ICs unless you have to. As in REALLY have to, like Snake/ROB have to. In almost all cases you want to outcamp your opponent and force them to approach you (lots of Ice Shot works wonders); ICs approach game is ****. When you DO need to, desynched SH Blizzard is more or less the all-purpose approach, try to avoid just running in. Squall also works if you need to approach in the air due to some kind of tactic directly stopping you on the ground (eg ROB + Laser), but in general you want to stick to the ground if you can since ICs are MUCH better from below than above. About the shielding issue, see my answer to 1. Anyway, remember that predictable isn't necessarily bad. Blizzard as an approach can be seen from a mile away, but it still ***** some characters nonetheless.
4. There is a pattern to everything she does. I believe there was a thread a while ago analysing her AI. Basically just play ICs heaps and you'll get a feel for her. If you want to dig up that thread then by all means do so.
5. Just mashing Back-Forth-A works for me. Apparently you're supposed to tilt-mash or something but whatever. Then again I may be thinking of the wrong desynch, a lot of them are similar haha.