Upvote:1
As you note in a comment, directive 2004/38/EC provides that EU citizens can stay for 3 months (and not 90 days) in another EU country without any requirement. You can indeed hop between countries. In practice, securing a residence permit as family member of an EU citizen in one of these countries can however makes things easier (no questions asked at border checks).
No, it doesn't, usually, but there are some caveats. EU citizens who made use of their right to free movement in another country (which might be your spouse's case before Brexit?) are covered even in their country of citizenship (that's the Surinder Singh route) and I believe Italy extends similar rights to its nationals anyway (maybe someone else can provide more details about that).
There is no predefined cooling-off period, the rule is really 3 months, not 3-months-in-a-6-month-period (old Schengen visitor rule) or 90-days-in-any-180-day-period (current Schengen visitor rule). On the other hand, if you leave for a day every three months, you could presumably be treated as a resident so what counts as enough time spent out of Italy is a bit of a gray area.
It's mostly unrelated to EU law and immigration/visa issues but if you end up spending about six months in Italy in 2021 (give or take, even without going over the 180-day threshold), you may very well be deemed a resident, e.g. for tax purposes. Other facts can also play a role (do you keep a home in the UK, etc.)