Those are proliferations. Leave them on the scape as long as possible so it can get some goid roots forming. I leave mine on there until the scape starts turning brown and dying. When you cut them off leave about 3 or 4 inches of the scape above and below the proliferation. This will help keep it stabe in the soil until it can get rooted. Sometimes mine will actually grow a scape and bloom while it's still on the scape.
Once planted they may wither down and you may think they have died but that's usually just the foliage and the roots are going strong. The foliage will start growing back.
That DL looks like STARTLE to me. Sure looks pretty. I have it, and it's usually so damaged by thrips and looks awful. Last year I finally got some decent blooms on it. I'm spraying for thrips this year.