...My info on obtaining permits for work answered your question #3 and is still valid, BUT if it's just the tax assessor, yes you have to let them in to inspect it too.
Your question is a bit unclear, but I am assuming you are doing some work/renovations on the property.
Usually ANY construction/renovation that requires wall removal, change/addition or moving of electric & plumbing, framing (yes framing of a new wall), venting, new windows, new roof, new water heaters/AC/furnaces etc...requires permits from the town/city. This is done to make sure all work is to code to protect not only you, but your neighbors as well.
If you are using a contractor to do the work, they are usually reponsible to obtain the permits needed.
If you are doing the work yourself, get to the town/city ASAP and find out what needs a permit, and what they need from you to apply for the permits.
Permits also protects YOU when you go to sell the place. Work done without formal approvals makes it ugly at best, and sometimes downright impossible to sell without YOU doing the necessary remediation to get the stuff up to code (translation $$$)
So, to answer
#1: if you are currently working on the house, and didn't file for a permit... if the town finds out about it, they will inspect it (could ne a neighbor, could be they just were driving around and saw the work being done).
#2 If you are doing work (with or without permits), yes you have to let them in to inspect, often all along the process.
#3 I think I answered that in the paragraph above
#4 Also answered in the above paragraphs.
Liberty Realty LLC