Eurail can be used in all those countries, but you will need to pay extra for the Eurostar between London and Paris, and for Paris-Belgium and Belgium-Amsterdam (if you take the fast train).

If you buy a flexipass ‘X days within a month’, you can use that many days within the validity period of the pass. If you buy a continuous pass (eg 15 days), you can use it any time within the validity period. 

As an example, the Eurostar will cost an extra €30 each way, on top of your pass.

And if you take the Thalys (the fastest train) between Paris-Brussels-Amsterdam-Paris, they will cost an extra €15-30 for each journey, but there are ways of avoiding the Thalys on some of this route.

You would also need to check there is availability for the trains you want.

You can use the pass in France, it's valid on SNCF's TGV services, Intercités, night trains and TERs. But not on the RER in Paris. Although most trains require seat reservations, except most TER. 

There are multiple ways to get cheaper in Belgium from France than via the (fastest) Thalys-option.

Or you go via Lille, Kortrijk/Gent to Brussels, or via Maubeuge, Charleroi to Brussels (slowest).

I'lld avoid Thalys too between Brussels and Amsterdam. There are hourly intercity trains withhout the need of paying extra reservations, only a tad bit slower.

Get your Eurostar reservations asap. They can sell out, and Alternatives are difficult. You can get the reservations via multiple ways, described here: 

Better check your trip details via a reliable planner. The rail planner app doesn't get a lot of updates and is often wrong. Best check www.bahn.com or the DB-navigator for up to date schedules all over Europe. Then count the travel days you need, and choose a pass befitting your needs :) 

THANK YOU, ralderton and BrendanDB!  Super helpful info.

Would you know of a company/individual that we could hire to organize and book all of the stops for us?

First I don't know a company but it is certainly doable to do it by yourself. You've got on this forum plenty of advice.

Of course people here will help you with specific questions.

The website seat61.com will help you a lot too.

Let me know if you've got any questions. I could help you a bit but I'm not qualified to organize a whole trip (although I often help my family)

@thibcabe, thank you kindly!
In that case, I will make an attempt at it.  Just all seems so complicated :)

I appreciate the help.