"That is me." Why is this incorrect and why should we use "that is I" please?

I think we can't say the pronoun "that" can't act as a subject: That is a keyboard.

Who is she? Here, too, "who" is not considered as the subject apparently! Is its reason the same as above?

What makes her happy?

But here "what" is considered as a subject!

I suppose the issue related to the problem is in the use of "to be" words, in the first two sentences above.

Do you agree?

All answers and comments are warmly welcome, I also would like to see StoneyB's answer. :-)

  • I would not say "this is me" is incorrect, I'd rather say it is colloquial and "this is I" is formal. – None Apr 9 '17 at 6:51
  • 1
    You might find the answers to that question on ELU inspiring. – None Apr 9 '17 at 7:05
  • 1
    You asked the same question on EnglishForums.com under the name of "KhoshtipMan", and were given accurate answers there. – BillJ Apr 9 '17 at 7:19