Can a school district official show up to my house unannounced to see if we reside at said address?

Yes, they can.  Public funding for schools is allocated based upon where the student resides.  In districts that allow open enrollment, the parents must complete the appropriate forms to enroll their student in a different district or specific school campus, which then enables the transfer of public funding for that student to the appropriate district and school campus.  It is illegal to secretly enroll the student using someone else's address (often grandparents or another relative).  Parents who do so may end up having to compensate the school district for the public cost of the student's education and will probably have to transfer their student back to a school that serves the area where the parents reside.