Most search boxes like that look for an exact match. That means that if what you are looking for isn’t the exact same as what you type, it won’t be found.

They’re not “search sites that are placed for your convenience”, they’re ads, or at least sold that way. Ad-blocker takes care of them completely. I can’t say how long I haven’t seen any.

Well, the one I just searched on, on my company’s website, couldn’t find a match for the exact name of my district. It didn’t take me outside of the website. Didn’t take me to any ads. I typed in the one-word name of my district and it didn’t find it.

I haven’t found them to be particularly useful.

They are usually just the standard search engine that comes with the website software. A good developer will replace it with a better one but this will cost money. It’s cheaper just to leave it but these search engines are often rubbish, as we know. Try using Google to search the site by using “search word search word site:http://www.website.com”. Wouldn’t work on an intranet though