***** NOT A TEACHER *****
What is your name?
(1) I believe that most grammar guides parse that sentence as:
Your name = the subject.
is = the main verb. (It is not an auxiliary in that sentence.)
what = interrogative pronoun serving as a complement (not "object").
(a) a complement "completes." If you said only "What is?," that would not be complete. When you add the
complement "what," then your listener/reader will understand. After linking verbs, grammar guides prefer the
word "complement," not "object." A complement is necessary to complete the meaning: He is intelligent/ nice, etc.
An object is not always necessary: I ate./ I ate an apple/ a donut, etc.
What did you do?
(2) I believe that you are 100% correct.
(a) The regular order is: You did do what?
(i) "what" is the object of the verb "do."
NOTES: Professor George Oliver Curme in Volume I of his masterpiece A Grammar of the English Language says that in "What are their names?," the subject is "What." I do not understand his reasoning. (His comments are on page 171.) I am trying to find out why he calls "what" the subject.
I found very helpful a Google book entitled Grammar for Teachers: A Guide to American English for Native and Non-Native Speakers (2008) by Ms. Andrea DeCapua. Parts of her book are available on Google "books." Check out her comments on page 252.
If you find out more information on how to parse "What is your name?," I hope that you will share it with all of us.