a) The subject of the portrait is wearing an elaborate headdress and colorful dress. She is probably from royalty.
b) The background is solid green. It is brought, and probably painted to show cheerfulness.
c) Although the subject looks odd, the artist wanted to bring out her inner beauty by showing her smiling. The artist wants us to view the the duchess as beautiful even though she is ugly on the outside. She would not have had this painting commissioned if she didn't want to showcase how she looked.
d) The subject is named duchess Elizabeth Lovell. She suffers from what would now be known as Hutchinson-Gilford progeria, a syndrome in which you age faster that you can grow. Although she is only in her late 20's, she looks much older. Despite her infirmity, she is confident in her looks. She is different in Renaissance society, not only because she looks different, but also because she is royalty, and royalty is supposed to be the epitome of perfection. Even so, she hired an artist to paint her to show her confidence.