Raspberry: yes. GPS: no.
none, not even a very expensive one.
There has happened much effort on how to track movement of a robot indoor for decades. Try purely mechanical tracking or, motion capture with a multi-camera setup or some kind of inertia system if you require accuracy better than 100 meters indoor. Just a rough guess, may be more uncertainty.
You do not need line of sight with the satellites. The signals pass through many types of roofing and flooring materials but they do get attenuated in the process. Depending on the actual attenuation and the sensitivity of the specific GPS receiver it may or may not work. My old Garmin Geko 201 struggles if not near a window in my bungalow whereas my newer GPS-equipped phones and tablet do work just about anywhere inside here.
Nope. A repeater can greatly improve reception. Any increase in accuracy? Only holds true for position of the receiveing end's antenna of the repeater: All GNSS devices indoor/receiving repeated signals will show the accurate position of mentioned particular repeater receiving antenna.
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