Are you aware that if you are on the train you don't need to take the carseat with you? And that the young boy can go for a walk with you in the train (or crawl) if he gets tired of sitting, that trains have toilets at every carriage or many carriages in the train and that many of those toilets come complete with baby changing tables, and that the trains go right into the centres of the cities you want to get to?
Are you aware that parking in den Haag (The Hague) is often as expensive as that in Amsterdam?
I see your concern for cheap parking hotels. I speak with a great deal of experience of traveling to all these places - I just got back from Luxembourg, by the way. The only place on your list that you may be able to park for free or very cheaply is at a remote Disneyland Paris area chain hotel. The remote hotels have large parkings, and run a bus to near Disneyland. The close in official Disneyland hotels charge around €20 per night parking, and Disneyland if you drive to it and park is even more for a day. At most hotels in the areas you are looking - Luxembourg, Düsseldorf, Ghent, Bruges, - you will probably be looking at between €15 to €25 a night. In Paris it may well be twice that. Really.
You won't be able to drive anywhere near the centre of Düsseldorf because a car rented in the Netherlands will not have an Umweltplakette - a green pollution control disk - on the windscreen and only the outer fringes of many German cities, including virtually all of the Ruhrgebiet where Düsseldorf is located is in the green zone. I happen to have just been looking at it on Wednesday.
Same thing with Paris. And Metz and Reims, where you may stop for a break because you can never get from Luxembourg to the vicinity of Paris in 2 hours. The French have a different pollution control system involving a yellow disk, and a car rented in the Netherlands won't have one.
Virtually all non-EU drivers driving in France (or named on the rental contract) need an International Drivers Permit in conjunction with their home Driver License. These must be obtained from the appropriate authorised authority in the country or state which issued your Driving Licence (spelled both ways because different countries spell it differently).
If you are concerned about the cost of parking, it will pale into insignificance when you see how much you will be paying for Autoroute tolls in France. Take along plenty of cash for that. And keep right as you approach the toll gates - the fast moving queues on the left are for cars with special transponders, and the ones in the centre generally are for chip and PIN credit cards.
You need to know that "national" or default speeds for each country - they are all different - for motorways, dual carriageways, single carriageways, in open land, in built up areas, and urban areas. In France the speed limit is different in dry and rain.
Speed cameras and red-light cameras are everywhere in Netherlands (they invented them), France, Germany, Luxembourg and Belgium. Only some are advertised, most are hidden. And expensive.
Does your carseat have ISOFIX fixings? That is the standard in Europe, but not in many other parts of the world. If not it is unlikely to be compatible.
This is not to dissuade you from driving - just to make you aware of all the tricky things you are setting yourself up for. In the case of your proposed trip - even with a 2 year old - going by train will be much much easier and probably overall much cheaper.
Do not rely on driving timings provided by Googlemaps. They are very optimistic, don't allow for any stops for food, walking or toilets, and make no allowance for traffic or (usually lots of) construction or detours. Use Viamichelin.com for better answers.
I drive or take trains depending on individual trips in Europe. For medical reasons I have to stop every 90 minutes and several times got stuck in traffic and couldn't.
