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). (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.)

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). (The notes are lacking a few sections, but they're nevertheless self-contained.)