If my first gift I gave my mother, after I began working, was china
and silverware, does that give me the first right of refusal when my
parents die?

A gift is a permanent transfer of ownership without expectation of any
repayment.  Officially your mother has the right to give the
china/silverware to anyone she wants in her will.  If she has no will
then the state (court) will decide who gets what.  If your
siblings/other relatives want you to have the china/silverware that
you gave her then there is little doubt that you will end up with it,
but since it is officially your mother's possession they have as much
right to it as you do should she pass away without leaving it in her
will.

Typically upon death without a will, possessions go to these people in this order:
(1) your spouse; (2) your child(ren); (3) your parents; or (4) your estate.

**Note that the spouse is usually entitled to everything before the
children are entitled to anything.