I guess the first one is the right one, simply because it follows common English grammar rules: Wh+aux+subject+verb+?

"Why should I lie?" is the correct one.

It depends upon the sentences in which you are going to use it .

"why should i lie" is a question and What i should lie... it's like a sentence so " should" behind subject " i " for example: what i should lie is telling my parents that i went out with my friends not my boyfriends. :D