I'm mostly fine with #1. That's K; long range Qs SHOULD be risky to follow up on; making them more so is reasonable.
The issue with #2 is not that it's more difficult- it's that it's flat impossible against an opponent playing competently (unless you have much more CDR than normal). Even if I E prior to using my R (starting its CD asap), have you in the middle or closer of my R (making you walk as far to get out of it as possible), use my wall to not only block your escape but also nudge you farther into my ult (ccing you briefly and gaining a small distance), fire my Q perfectly such that you cannot dodge it, and use my E immediately as it comes off CD... you should still not receive chilled damage if you simply walked away from me. Keep in mind that for the scenario I just listed, the enemy player had to have done a TON of things wrong. They have to be within 650 range, have no gapcloser or mobility available to them, have no jungler/assistance nearby, AND have no ability to all-in me. This is post- level 8 Anivia, too- you really can't do this in a meaningful way until then, because you simply don't have the points in your skills for this combo to work. If an enemy mage or assassin is THAT close to you without being able to fight you, at that point in the game... they deserve to die, assuming you play optimally. I feel it's very likely that this change will make that not possible to do unless the enemy just meanders around in your ult for a few seconds so your E can come back up.
Point #3 is silly. It's not a case of "oh i outplayed your E damage," it's a case of "wtf my targeted skill isn't doing the damage it should be doing." As the recipient of the E, you should ALREADY have been outplayed (in terms of your ability to avoid the bonus damage) if I apply chill to you and follow up with an immediate E. After all, I have a pretty moderate range, and most likely your champ has more mobility than I do. The only champs that DON'T have either a dash, significant speed boost, or higher base move speed than Anivia are Ashe, Jinx, Kog'maw, Soraka, and Zyra. Four are longer ranged than you, and the fifth doesn't want to damage you anyway. You didn't HAVE to get that close to me; you chose to.
Now, if they changed things around somewhat- say, it calculates whether it deals double damage or not on cast, rather than on impact, but also decreased chill duration a small additional amount (~.25 seconds, maybe as much as .5), I wouldn't have any problems with that. It's still a net nerf- ~1.25-1.5 seconds less slow time as well as ~.5 seconds less chill vulnerability time, but it's not quite as ridiculous.
I'd also be alright with several other changes. Maybe you make the attack speed debuff on the R only 15%; maybe you up the mana cost on the wall a little (it was decreased some time ago); maybe you make Fear, Taunt, and Charm interrupt her R (they actually don't atm). Point is, there are lots of places that you could make nerfs that I think make more sense. This nerf heavily hits some of Anivia's more difficult/skillful to pull off combos in particular, which really isn't where she should be nerfed imo.
+ 
. No, no, no. Yasuo needs to be the one that dominates and owns mid!
[slayer-jinx-wink]
Last Emperor of Shurima, Ruler of the Desert, Owner of the Middle lane, and most importantly: The wrecker of Yasuo's shit.
This guy needs no introduction.
