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Are you sure you want to spend only 2 nights in each place?

That gives you only one full day to tour/visit and a few hours on your arrival departure days.

Doing that on a tour that has door to door transportation and a 'never lost' guide that pace can work. As a DIY project taking rail, it becomes an endurance test, not a 'family vacation'.

If your flights aren’t booked, I suggest that you have a week in London and a week in Paris. If your return flight is booked, have 6 nights each in London and Paris and the last 2 nights in Brussels.

Sorry, but 2 nights = 1 day and you aren’t going to get to see anything other than train stations or airports.

Should I get a Flex Rail pass for 7 days? Is 7 days enough?

A1) Not a great train guy. Use to be individual tickets were a better way than the passes, but with your schedule and the passage of time, cant help.

Should I get a SIM card in order to see the city mapper?

A2) Or a US carrier with international service like AT&T or TMobile (easier from my point of view)

Besides the 3 places we'd like to stay for 2 nights, which other(s)
would you suggest to stay for 2 nights?

A3) Read the other posts above; and soon to follow.

One more post to state what others have said, this is not realistic. You don't have to limit yourself to just London and Paris, but 7 cities is way too many for a fortnight. London, Paris, Brussels and Amsterdam on the other hand are easy to combine.

Thank You everyone for the prompt responses! I'm grateful for all of your replies! I will read carefully each of your feedback and adjust my plan.

You DON'T need a SIM card to see a city map or get directions.
I've used Here WeGo ( https://www.here.com/applications/wego ) on an old phone without a Sim, and it's worked well around the world. You just need to download the city maps of London, Paris, etc. in advance (all free) and when you get there your phone's GPS will show where you are on the map.

Thank You so much again! I just downloaded the app!

OK, now we are down to 4 :-) ....... London, Paris, Brussels and Amsterdam.

Should we get a 3 day pass for the family of 3 for traveling between the above? And buy bus pass at each major city?

Thank You again everyone!

London to Paris is done by taking the Eurostar which gets you there in 2.75 hours and takes you under the English Channel.

Paris to Amsterdam would be done by train in just over 3 hours.
Then you'd backtrack to Brussels by train in order to fly home from there.

Should we get a 3 day pass for the family of 3 for traveling between
the above?

No, buy single tickets.

Scale back your plans, you are trying to do way too much in only two weeks.

2 nights in cities like London and Paris are not enough. Better to plan 4-5 days in London and Paris. Brussels one night is no big deal. Of course, you should go to Brugge, Belgium.

Consider London, Paris, Brussels, Brugges, Amsterdam for your two weeks.

Yes! I now plan for 4 days in London and Paris, then the rest will be wandering around Amsterdam, Brussels and Bruges.
I'm so thankful to all of you who have taken the time to help us with our family trip! Much appreciated!