That was normal behavior when I had AIS, nothing to worry over.
As to question 1, yes, because you reset the modem, it is being recognized as a new network.
As for question 2, These 3 options represent how communication will be allowed to and from your PC, Home being the loosest, Public being the strictest. I always left mine on Work/Medium, even at home, just as an extra precaution, but you should not need the Public setting unless you are on something like a public WiFi hotspot.