Is GPS data signed and timestamped by the satellite?
No, GPS is not signed. I think there may be in future some signed signals. Maybe it is used on military part of GPS.
GPS spoofing is a well know problem, and problems are usually "solved" in hardware. If signal strength on some satellites changes quickly: do no trust it. Multiple antenna (on extreme of lorry/ship) and comparing strength of signal helps. Some directional antenna helps to know that signal expected from a satellite come from where it is expected. And often some gyroscope, compass, or and tracking, to check plausibility of data.
Note: fishing boat uses it to decoy own position (in protected area/economic exclusive zones). They causes a lot of troubles to other ships and boats (and sometime to ground equipment).