Plane tickets advice

$700 roundtrip JFK to FCO and then CDG to JFK during the high season sounds like a good price. I would snag it.
I can not see the price going lower from New York with all the demand to Rome and from Paris.

There's definitely nothing magic about buying any particular number of weeks before the departure date. Once I know my European gateways, I start watching fares, and I buy if I find what looks like a good deal. My origin airport is relatively low-cost (though not as cheap as NYC), and the best I've found for a summerish trip was around $540 into Nice and home from London. That's a cheaper combination from DC, usually, than Rome/Paris, so the $700 international multi-city ticket you've found sounds like a relatively good price to me. I wouldn't expect that fare to be much lower even if there's a short-term price drop, and I would expect a considerably higher fare over time.

You have to decide how comfortable you are with the risk that something will go wrong with the domestic feeder flight on the outbound trip. That would probably be crushingly costly. I'd be considering flying to NYC a day early and paying for a cheap motel somewhere. But I'm very risk-averse, and I'm retired so I don't have to worry about an extra day away from home.

What are your travel dates and where are you flying from? You may be looking too early. I fly open jaws and not separate tickets as that is too risky.

$700 is good. There’s no magic number of months to wait. I’ve bought at 11 months out and gotten excellent deals, but not always. It just depends. You can’t crack the algorithm, just buy when you see a price you are comfortable with and are sure of your dates/cities.

In general I keep it under $900, often $800 for summer travel. Sometimes less, but that depends on destination.

$700 is good but with the extra $300 to get you to NY and issues with an unprotected connection (buying 2 separate tickets on one route) I'd have to think long and hard. IF you have flights all on one ticket then the airline is responsible for getting you to your destination. If you are flying on different tickets and have delays on your originating flight that cause you to miss your International leg you will likely have to pay for new tickets for your group.

Will you need to fly to NY the day ahead, which would mean an overnight in a hotel? How many flights from your home airport to NY per day? Are you flying in winter or in a busy time of year (holidays?).

You'll need to leave yourself a decent amount of time for a connection in NY so keep that in mind when you are looking.

I would suggest setting up the tracking feature on Google Flights, using open jar flights. Set up both options- originating from JFK, and from your home city. Then follow the price fluctuations.
It's amazing what prices can show up, and how options/prices change just throughout the day. And if prices go up, they usually come back down a week later, [as long as you're several months out.]
Safe travels.