I have a paid Google CSE.  So far the implementation went super well...just past the code into my mydomain.com/search/ and I am good to go.  However, there are a couple questions I have.

1) How do I create just a search box on the top of each page to use this search?  The only solution I could find so far does a search for my site but on google's domain.  I just need to know what parameters to send it.

2) Is there a way to have my CSE page update the URL with the variables for the search?  Essentially, submit the query using GET? I want our tech support department to be able to copy/paste the search query URL to customers who have questions on something.

Thanks