Damage over time and item rarity support

Let's say I hit a monster, and that hit applies both bleed and corrupting blood from corrupting fever to the mob. When the mob dies, he still has the bleed and the other dot on him.

Corrupting fever was supported by item rarity support.

According to a comment by Mark_GGG, in case of multiple dots, the oldest one is considered the one that kills.

So in the scenario described above, is the bleed or the corrupting blood "first" and do I get the bonus from item rarity support or not?

Also, is the outcome for item rarity any different if I hit the mob multiple times before it dies, and some bleeds are weaker than the previous ones?

Rarity support gem only applies if attached skill scores a kill. Only oldest dot can score a kill. Then, all stacks of bleed you inflict are present on target and expire individually, but only the strongest stack deals damage (more with crimson dance), and thus can possibly score a kill.

So, to make iir gem work with bleed you have to make sure your strongest dps bleed comes from the attached skill. Same logic for corrupted blood, except it stacks with itself damage-wise

I don't understand what you were trying to say.

Which one gets the kill then if both are from the same hit - the corrupting blood or the bleed?

If your question was specifically about corrupting fever - it is a spell which does it's own damage (in the form of corrupting blood), so it is not considered attached to the skill link you are doing a hit with. Instead corrupting fever needs to be supported by its own support gems.

So, when enemy is killed by DoT damage, the kill is awarded to the owner of whatever dot which was applied the earliest (among still active dots at the moment of kill).

In your case corrupting blood and bleed have different durations so even if we knew the dot application order on a hit, it would be impossible to figure out which dot was the oldest at the moment of death, so unfortunately the only way to be sure is to support both corrupting fever and your main skill with their own item rarity gems.

Maybe someone else here has better knowledge about this than me.