So does this ruling apply to the public or only to government?
For example, could I legally collect Elizabeth Warren's "inadvertently shed" DNA and have it tested to find out if she really has a Native American ancestor?
Well, the case is about "a case that examines whether police should be allowed to collect and analyze 'inadvertently shed' DNA without a warrant or consent" which has nothing to do with any of those you are talking about. It is very specific on whether "police" or law enforcement can do that. Not about unspecific circumstances.