Sorry for replying on an older question, but I would like to clarify the last question.
You use a "get" method for your form. When the name of your input-field is "g", it will make a URL like this:
https://www.google.com/search?g=[value from input-field]
But when you search with google, you notice the following URL:
https://www.google.nl/search?q=google+search+bar
Google uses the "q" Querystring variable as it's search-query. Therefor, renaming your field from "g" to "q" solved the problem.