3 questions related to incinerate (for experts)

Let's assume I use "incinerate" as my main dmg skill and "added cold damage" as a support gem.

Now for the questions:

1.)
I know that incinerate is unable to crit but together with assasins mark it is possible to crit (10% on lvl 21 curse gem) for incinerate. What about the "annihilation" node and it's "50% increased critical strike chance for spells"? Will this bring the crit chance up to 15% together with assasins mark?

2.)
Will the node "celestial punishment" affect the crit chance too? I mean 5% chance to freeze, shock and ingite should be the same as 5% crit chance.

3.)
The tree node "frostwalker" offers 18% increased cold damage and helps a bit freezing mobs up to lvl 75. What if I use a "pyre" ring? It will convert 100% of cold dmg do fire dmg. Of course I will loose the freezing part of incinerate but is the increasing dmg part of the "frostwalker" node still working or will this become completely worthless because now I only deal fire dmg?

Thanks in advance!
1. no
2. no, chance to apply a given status ailment is a different from crit. But it's an effective node with incinerate as it hits very often.
3. Fire damage will benefit only from fire damage increases, cold damages will benefit from both fire and cold damage increases. So it won't work on Incinerate but it will work with Added Cold Damage, and this support will also get boosted by your fire damage increases.
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