I have a non immigrant O visa based on marriage. My wife will be giving birth at the time when my visa expires.

 

I want to do a 60 day extension after that and a 1 year extension after that.

 

Question: when I do my 60 day extension, does my wife need to come with me? Or can I do it by myself?

You wife will need to be with you when your apply for the 60 day extension. You can do the extension of your 90 day entry up to a couple of weeks early.