Well, as with any other ad on your site, it all depends on your content and your placement. If your users need the search function to find anything, they may be more likely to use google's search. After that, it all depends on what they are searching for. My advice would be to try it out and see. If you feel like its taking up valuable space for little return, remove it and put in something better.
I am pretty sure you can't have Google as well as another search function. Make sure you read the TOS before you implenment it.
I just skimmed the Google AdSense terms but did not see anything like this specified. Where did you read this or hear about this? Can you show me where you found this?