Write to serpcheckerext@gmail.com and you will get an answer from the person who wrote the extension. Before you do, the questions below are the ones that come up most, and two of them are things people reasonably expect to work that do not.
Two things cause this. If you gave google.com permission to see your location at some point, Chrome sends your real position with every search, and for a city or a country it used to win over the location you picked. Version 2.0 removes that header, so update to the latest version first. If it persists, check that the switch at the top of the panel is on and that the page you are looking at is Google Search rather than Maps.
Countries, regions and cities come from a list built into the extension, 114,000 of them, and work offline. A full street address is different: it has to be looked up, and the extension asks your permission the first time. The address service is built on OpenStreetMap, which is excellent in some places and thin in others, so a new building or a complex known by its marketing name may be missing. Typing the street and number rather than the building name usually finds it.
It is how precisely you claim to know where you are, not an area to search in. We measured it against live Google: from a hundred metres to a hundred and fifty kilometres the results are the same, in a capital city and in a village of 1,400 people alike. Only once the circle covers a whole region does Google stop treating you as being in that town. Leave it exact unless you want the regional picture.
No, and nothing that claims otherwise should be believed. The extension changes what Google believes your location to be, by adding Google's own location parameters to the searches your browser makes. Your IP address is untouched. For anonymity, use a VPN; the two work together.
Nothing, unless you switch on the optional usage statistics, and the extension asks once. Your Google searches are never included and cannot be: the extension adds parameters to requests your browser makes and never receives the traffic. The full detail is in the privacy policy.
Tell us what happened, what you expected, and which version you are on, which is at the bottom of the extension's settings screen. serpcheckerext@gmail.com.