Need Opinions For Our Holiday - First time taking the trains in Europe

Very good choice to travel by train ! Let me help you a bit :

There are no “Eurail train stations”, you use the same stations and trains as people with regular tickets. The Eurail Pass is basically a ticket covering all European trains (with a few little exceptions like Paris RER or Ouigo low-cost high-speed trains).

I’d recommend reading these 2 links first :

https://www.seat61.com/how-to-use-a-eurail-pass.htm

https://www.seat61.com/interrail-and-eurail-reservations.htm

Most of the time you simply log the train to the Rail Planner app before baording and hop on, it’s as easy as that. As you can see reservations are mandatory on fast/high-speed trains in France (extra costs), you have to book these journeys in advance limiting your flexibility. However there are always slower alternatives. When is your trip planned ?

There is no luggage limit as long as you can carry it yourself. You shouldn’t need a large suitcase for 13 days anyway.

How many days per country is frankly up to you. I’d set a base in Belgium and visit other cities on day trips (it’s a small country) I also wouldn’t spend more than a night or two in Luxembourg.

You should calculate how many pass days you’re likely to use. It’s not always worth using a pass day : for example Belgian trains are very cheap, especially if you’re under 26 y.o.

Last thing : I would not rent a car. Traffic is horrible, parking is scarce and expensive → stressful experience.

Feel free to ask for more advice.

Honestly, based on your rough plans. I think you'lld better not invest in a pass, unless you really value the freedom of hopping on train to train. 

The Belgian railways will probably have an unlimited pass for the summer for a very modest sum (+/- 50 EUR, for taking unlimited trains in Belgium, available on the app of the Belgian railways) and public transport in Luxemburg is totally free! 

The Netherlands is a bit more expensive, but if you travel off peak and buy a bit in advance, quite affordable.

In the BeNeLux you can get almost everywhere by train and the combo with other public transport, no need to rent a car. Many cities also actively discourage cars, very not worth it to rent one for your travel plans. 

France (TGV and Eurostar (ex Thalys)) can be cheap, when booked in advance. The only downside of booking in advance, is that you'll have a bit less flexibility. So keep an eye on tickets, best booking window is about 3-4 months in advance and compare with the cost of a pass and SNCF's/Eurostar dreaded reservation costs :). Thalys can get expensive, but easily avoided by alternative routes with one or two extra changes (probably via Lille). 

To reply on other questions:

High speed trains (mainly TGV and Eurostar) have mandatory reservations that bind you at one specific train, so they can sell out. All the other regional and local trains in Belgium, the Netherlands and Luxembourg are non-reservable. They go at least hourly, and between the biggest cities you have trains +/- half hourly, sometimes even quarterly or 10 min intervals. They have plenty of capacity, peak times can be a bit busier or towards popular destinations (like the coast on nice summer days). But 99% of the time, you'll find free seats :) Buy a ticket (or register it in the app) and good to board! 

Stations are plentiful and well connected by bus or tram, or the underground in the biggest cities, rental/sharing bikes and steps are usually also available,... And stations are not far from the things worth seeing anyway.

Pass days. The concept is simple: On an active pass day, you can use as much trains as you want, as long as you register them in the app. 2 or 20 trains, doesn't matter. You can do the same route all over, or go for a very long train ride. Some trains have mandatory reservations (TGV, Eurostar, night trains) and you'll need both a seat reservation and a valid pass (active travel day, and registered train that you're taking). It's quite straightforward when using it. No worries. See it as a train travel diary. 

That pass still exists (cheaper for under 26). There are also weekend return tickets. And more.

I can also recommend Leuven and Gent. More so than Bruges IMO.