There is nothing called "family ticket" for Interrail/Eurail.
Children under the age of 4 don't need a pass unless the need a seat of their own.
Children who have turned 4 but not 12, and children under 4 who need a seat of their own, need a free children's pass. Such passes have to be ordered together with the adult passes.
It the child's age on the first travel day that matters.
With a mobile pass you don't need to decide what date you want to travel when you buy the pass. You need to activate the pass no later than 11 months after purchase. If you buy a paper pass you must decide the start date of the validity of the pass at purchase.
I see you live in the UK, but is it correct that you wouldn't go by train from the UK to Lyon?
Mit einem Interrail-Pass könnt ihr nicht innerhalb eures eigenen Landes fahren, außer an einem An- und Abreisetag.
Ihr müsstet auch für die Strecke von Lyon nach Casses und von Casses nach Nizza die regulären Zugverbindungen kaufen, zumindestens wenn ihr dort auch Aufenthalte plant und nicht einfach nur durchfahren wollt.
Ich empfehle euch die französischen Zwischenhalte auszulassen und mit Umstiegen direkt von Lyon nach Savona zu fahren (Etwa 8 Stunden Fahrt). Dann reicht euch für die Strecke in Italien ein 5 Tage Ticket, sofern eure Kinder nicht älter als 11 sind.
Beachtet auch, dass ihr die Nutzung von bestimmten Schnellzügen zusätzliche Reservierungen kaufen müsste.
Viel Spaß beim Trip :)
For mobile passes, you can activate the pass whenever you want in the next 11 months, so July/August should be perfectly fine. Then you can freely chose the travel days within the timeframe after activation (1 month for the 7 day flexi pass). For paper passes you chose the activation date when purchasing (or possibly you have to activate them later, I’m not sure)
You are indeed doing 7 trips, so 7 travel days within 1 month is enough. Note these travel days should be within 31 days (when activating the pass in July). If your children are under 12, you can order free passes for them too together with your own.
Most of the journeys you mention are however relatively short regional train rides, so you might be cheaper off with buying full tickets. For example, all these tickets can be bought until shortly before departure from Trenitalia ticket machines or their own website:
- Savona-Genova: €4,90
- Genova-La Spezia: € 8,40 (RE)/€9.30 (RV)
- La Spezia-Forte Marni: €6
- Forte Marni-Pisa: € 4,80
There are different ways to get from Lyon to Cassis and from Cassis to Nice, but I think that also including those don’t make buying a pass worthwhile for just these trips. It might be different depending on the age of your children (kids under 12 get free passes when purchased together with an adult, but only 50% off ticket prices of TER's and Italian regional trains - so include that in your calculations).
You could also combine full tickets with a smaller pass, or go for the pass anyways if you want to do larger detours on those trips (or a day trip to somewhere). I generally use an excel sheet to determine what pass I need - including the costs of full-price tickets - and then I often end up getting the 10-day pass, which has the added advantage that you can spread the travel days over two months (generally 61 days). The price difference with the 7 day pass is not that big, and it gives me a lot of flexibility to fill in trips later on.