1)
No. A new urn is prepared for the deceased.
2)
The person in charge(in the crematorium) usually picks a bone with explaining what part of born it is (e.g. This is a femur. This is a rib.), and passes it to a mourner. He/She places it in the urn and then the next mourner recieves the next bone from the person(= the person in charge).
3)
It IS done indivisually. The situation "bones of two dead people are in the same room" never happens.
4)
The person in charge usally says "You can pick bones by yourselves. Or, I can help to pick them and pass you. Which do you like?", but I've never encountered a case of doing it only by mourners. (I have many relatives. I've attended those ceremonies ten times or so.)
5)
Not "by family members", but "by the chief mourner(moshu)".
Yes, it is done at the end before the skull. As you might know, the adam's apple is called "nodobotoke" in Japanese, meaning "throat Buddha". This name is from the appearance of the bone. They say it seems like a sitting Budda in meditation. So, the adam's apple is placed in the urn in the right sitting form, and then after the urn is covered, the person in charge marks up the cover to show where the front of "the Budda" is. (When the urn is placed in the little chamber under the headstone, the mark is faced to the front.)
6)
The bones are as the body is in the coffin, not being mixed up in a tray.
P.S I am planning for the guest children to do the bone picking, then the parent's children who are frightened to see their parent's bones picked will force themselves to pick the adam's apple bone together and place in urn before a friend says they will finish it off by placing the pieces of the skull in the urn
That order of bone picking is not common.