A few things:
1. Yes the LinuxLite search page now shows country correctly for the US.
2. Just an idea to consider: have you thought of changing the bit that says "Linux Lite Support" to "visit Linux Lite Support"? Either that, or have the "Linux Lite Support" link appear in the line ABOVE the gmail, image links etc. Otherwise as it stands now there is a small chance that people will expect the search box to limit the search to what appears directly above it, which would be Linux Lite Support.
3. Separately, regarding only the proposed idea of a link to search in forum profile, it's an interesting idea but I personally would find it confusing. Unless the link was labelled just "Google" or something like that, and even then, I'd be under the impression that clicking on it would search only the support forums and nothing else. If that is the function of the proposed link in forum profile (i.e. a Google search within the LL forum site only) that would be fine, but if it goes to a general google search the exact same as the browser home page for 3.8, then it would be confusing to me.