You can always use Google search feature with your adsense account on any of your sites/blogs. The only thing to avoid warnings (or account suspension) from Google is, never click on the ads which you get after searching through their search feature. Also, you should avoid using Google search feature for yourself on that very site too,as,some webmasters have received warnings from Google for that reason too.
Its always better to be on the safer side with Google and they will always be a good source of income without the hassle of pop-ups and undue redirections to the visitors.
P.S. Its always your choice to include your site's name in Google search button or not, so, you can go without adding your site name too.
