"My wife is now giving birth to our first baby on an operating table!"
Translation:제 아내는 지금 수술대에 우리 첫번째 아기를 낳고 있어요!
When you want to express, where the action took place (in the shop, on the operating table, at home etc.), you can use either -에 or -에서. Which one to use depends on the verb. Most often you should use -에서. I know only two actions, that require 에: these are 있다 (to be) and 서다 (to stand). In most of the grammar books they explain, that -에 is for static verbs, -에서 is for verbs with some action.
So in this case, as in our sentence, with the verb 낳다, it feels like even according to the grammar books it should be -에서.
You use 먼저 when you want to compare something with something eles, and 첫번째 has exact same meaning as "first".