I think all three questions are answered by Proposition (1.5) in Chapter PARTL ("Partially symmetric functions") in [*The LLPT Notes* (by Dan Laksov, Alain Lascoux, Piotr Pragacz and Anders Thorup)][1]. (Disclaimer: I haven't read the notes far enough to tell whether the proof is complete; the notes are somewhat of a construction area in some places.)


  [1]: http://www.math.ku.dk/~thorup/notes/sympol.pdf