Just to be clear, when a doomed time line is erased after a certain amount of time do the people from that time line get to "live" in the dream bubbles or are they just erased from existence completely? Also who erases the doomed timelines or does it happen naturally somehow?

They are completely erased, that’s why we never seen any Gamzee’s in the dream bubbles. And they’re erased by Paradox Space.