1. It would be okay, Google has an official custom search API anybody can use, but the free version only allows for 100 searches a day, after that it’s 5$ per 1000 searches.
2. This is a terrible idea. People will not want to use it. The browser is a tool that has been evolving for many years, and the browser vendors have slowly improved their interface and handling to be as easy and convenient as possible for all users. I can assure you that you will not make something that’s better to use by embedding a Google search browser box thingy where the game should be. It would be tiny and hard to navigate, and absolutely nobody will use it.
I think the real issue here is that you personally don’t have good computer knowledge and habits yourself, and you project that onto others by assuming they have the same difficulties that you do.
Did for for instance consider that you can just resize the browser window with the Kongregate chat to cover half of your screen, and then open a new browser window with Google, that covers the other half of the screen? That would be exactly what you are describing, you would be able to chat and read the replies, while searching around, without having to switch back and forth between the windows.
See how easy it is: