The first sentence is correct.
Why did you go there puts the subject (you) of the sentence in the past and the verbs (go and do) describe the actions of the subject (you) which was already described to be in the past.
Techincally if your friend was being consistent their argument would have been Why did you went there and did that? Which is very incorrect....
"May I know" is found in British English, but sounds very formal and old-fashioned.
The GloWbE corpus shows it appears far more in Singapore and Malaysian sources, and then in Indian and Filipino ones, than anywhere else.
The simplest expression would be
She is more beautiful than me.
(The final pronoun is being not heading a clause so it can take the default "me" form)
However it is equally correct to say
... than I. (formal, slightly old fashioned)
... than I am. (correct but the repetition of the verb is not needed)
There are times when the full comparison ...
Please tell me once eat lunch or Please tell me once ate lunch or Please tell me once have eaten lunch
Well, none of them are correct. You wrote
I want my friend to call me once he is finished his lunch.
One way to say this is
I want you to call me when you have finished your lunch.
Please call me when you finish your lunch.
We can also say