New to Kayn, is sunfire cape good in him?

You dont really need it. His Q is more than enough AoE on a short CD and does extra damage to monsters.

If you need an armor item for Rhaast, Dead Mans Plate is awesome.

Sunfire cape is just not an efficient item on him. You get Sunfire on tanky tanks that need extra help with wave clear (and a bit of AOE dmg). Kayn has amazing wave clear with a simple W-Q. If you're building him tanky, a deadman's a better option given that movement speed and and an on-hit dmg / slow proc is something kayn likes.

Is everyone ignoring the fact that you would most likely be jungling as Kayn, so you will generally always go for Cinderhulk over Sunfire?

Cinderhulk is fine on Red Kayn, but probably won't make much sense on Blue unless you want to try and break League with another tank assassin (the community loves those).

Warrior is probably "better" because Kayn is generally more useful as a bruiser than a tank, but sometimes you need to be a tank.... go for it.

As for your question about the Immolate passive working while you are "inside" your target ... I can't recall off of the top of my head, but my gut says that it does work.