You're Plat, flying C on AA metal to Zurich and continuing on Swiss (AA codeshare flight #) on to FCO. You are not a member.

On the outbound:
Use of the clubs domestically on the day of international travel.
Arriving in Zurich - I have never been there. I don't know if AA has an arrivals lounge or if they contract some other one for arrivals. If it's a yes, you will be able to use arrivals lounge. You will not be able to use any other OW lounge. That is only allowed if you were continuing on a OW marketed and operated flight. Codeshares on other partners don't count. Swiss is not a OW partner, yet.
Again, I will say this, can it happen? yes, but it's not a given perk.

On the return.
FCO - no perks, no lounges.
ZRH - Any OW lounge is fine.
Domestic - if making connection and there is a club available, again you're in.