These are all really good questions, no worries! While she would technically be accepted as a member of her new Court, we would like to leave how she is treated down to the PCs in the game which should allow you more freedom in building CR based off what other characters do (especially if Lucrezia has been at odds with her new Court for a while!). It would be a very interesting tangent to take!

And she could definitely keep all her boons/powers, yes.

EDIT: Along with her status! Whoops, forgot to add that.