I would suggest removing all of the old backing ring if you are making a new weld.  This can be done fairly easy by machining or grinding.  I don't see a benefit to leaving part of it inside the pipe, since it could dislodge and travel down the pipe where it might lodge in a valve or go through a pump or other equipment.  The codes don't address this level of detail.

Chill rings are normally used in structural pipe applications and may be lft in place or not, depending on the Engineer's design.
Often, if left in place and the pipe is big enough to allow access to the inside, the outer edges of the chill ring are seal welded to help prevent corrosion due to a break in the coating.
Chill rings are not typically used in fluid carrying pipes due to turbulence, but they can be Engineered in if desired.
As far as the codes are concerned, they just want you to be sure that if a welder has qualified with backing, then backing must be included in the weld joint, whether it's left in place or not.
Of course it's been a while since I worked with ASME. I'm sure if I'm incorrect someone will add a correction.

Tim