好问题！good question! (1)为何=为什么=为啥 ，just use in different occasion. 为何 official, 为什么，common, 为啥 colloquial e.g. : why to do so=为何这样做/为什么这样做/为啥这样做 凭什么 include some emotion,like "angry" 凭什么=why + can't /should not usually,“凭什么”is not to ask a question, just to express "strong dissatisfaction" e.g. :凭什么这样做=you should not to do so. I'm angry. (2)"好" has two meaning: good / okay、all right “ 好了” means "done"/"complete、finish to do something" "好的“ means "okay,all right", can be instead of "好” “好吧” ≈“好的”，but "好吧“includes some emotion, like "however,all right" 希望回答的这些，能对你有所帮助。

一、 为什么 ：is natual, common formal & colloquial. 为啥 ：colloquial, litera, social-network(between best friends). 为何 ：written, formal, social-network(between best friends). 凭什么 ：When you feel angry, you can say '凭什么...'; Also '为什么' is also okay here. 二、 好 means okay, alright. It is very casual, colloqual. I would use '好' when responding a best friend's 'invitation, ask'. 好了 means "something has just been done" or 'You have just finish doing something.' 好的 means "okay,alright". I feel '好的' is more friendly than '好'. 好吧 ：When you don't want to do something but you have to, you can respond with '好吧'. But don't say this to your bosses or acquaintances.

1）为什么=为啥=why different in tone I guess. People from different region talk differently. 為何 can be "why' or "for(為) what(何)" 凭什么 is kinda rude I think. Feels like....."Based on what?!" literally. 2）好= good、nice、OK 好了= It's done. 好的=OK (when agree with others) Yes (answering to order) 好吧=alright. EX: Alright we can do this later.