Q.1. How to Add Google Search in a Beans child themes without plugin? Q.2. How to add secondry Nav bar in beans themes?
If you want to embed your Google Search, you could just add the code provide by Google in a widget or add it anywhere on your page using Beans hook. Is there a specific area where you would like to add your google search?
Regarding the secondary nav, kindly create an discussion for that as it is not related to the Google search discussion.