When adding a Google Custom Search box to my site, does that help SEO, and does it influence regular Google web results? If not, does it provide me with data to help me optimize my website?
Using Google custom search will not affect your pages' rankings. If you're looking for data that will help optimize your website sign up for a Google Webmaster Tools account. It offers a bunch of useful information for how Google sees your website and even allows Google to communicate problems to you.
it seems that this may have an impact, as explained here:
Is Google Custom Search Influencing Google Web Search?
There are many articles that support the rumor of Google custom search engine helps your SEO. I am not sure if it doesnt make any difference on the SEO or if it improves the revenue but its for sure its the best promo message for Google custom search. If there is anyone who has checked seo impact from google custom search usage in his site, please share your story :)