Are all natural diamonds made of organic carbon material?
Diamond isn't made of organic C at all. Organic matter would rather become oil, gas, coal or dissolve entirely.

C itself isn't very common in earth's mantle, but subducted eclogites and peridotites can lead to the needed C-accumulation.

But also meteorite impacts can lead to the genesis of so called micro-diamonds due to the very short lasting but extreme conditions. 

Diamond's genesis takes place in depths from 150 up to 700km and at temperatures about 1200-1400°C.
Those conditions were present in the upper mantle during the archaeum and hadeum, that's why mining takes places mainly in older crustal zones.

You are right that volcanic activity brings diamond up to the crust and surface of the earth. This process is thought to take only some minutes... This is a very quick process so that the diamond crystal structure won't change back to graphite and it's also the reason many diamond deposits can be found in sediments.

I think that should answer your question.
I don't know about the diamond exoplanets, though I've heard of it.. I'd say the process that leads to this is pretty unclear, but perhaps someone has clearer thoughts or information on this.