Arriving from non Schengen country

We are driving to Italy at the beginning of April for 12 weeks. As we live in the Channel Islands (which are non Schengen) do we need to register with the police on arrival? Also does our time in France count towards our 90 days as they are a Schengen country?

We are also confused by the advice online about the requirment for snow tyres and/or chains. We will be driving down the coast from Genoa to Pisa and will arrive around 8th April.