Day trip to Castle Jehay near Amay Belgium

Sorry to say that according the castle’s website the castle is undergoing a restoration and the interior not open to public. http://www.provincedeliege.be/en/chateaudejehay

But if you still want to go you can certainly use the train, but as Amaj is a small place likely there are no taxi’s waiting, presume you will have to arrange one before arriving there. For planning the train journey: http://www.belgianrail.be/

Have little experience with driving around Brussels, but is notorious for traffic jams. Think getting out of Brussels is not directly a problem, but bringing back the car is better somewhere outside Brussels (like Leuven) and take the train from there will be easier.

Google StreetView gives a long row of taxis waiting in front of the railway station in Huy, a few kilometres from Amay.
For the return trip it's good to know there is a bus connection to Amay and Huy. It is bus 85 run by TEC, and has a stop close to the château. You'll find the schedule on their website by filling out »85« under »Ligne« and selecting »85 Liège ... Amay - Huy«.

The Castle Jehay is a heritage sight for my family... hence the main purpose to visit it, even though it is closed. I am currently thinking that we can take the train to the Brussels airport, then rent a car and drive out to Amay, Huy or thereabouts, spend the night, see Castle Jehay, then back to where? Can we drop the car off at Leuven or drive on to Ghent and join the RS tour there?

Of course, you can drop off a rental car at whatever office you selected online when you booked the car rental. The huge international companies all rent cars in Belgium, although there may not be enough automatic transmissions for you. You should read our host's travel hints (top left blue box) about driving in Europe. And I can tell you that Brussels-Gent is a high-traffic area, just like much of the U.S.A. (I'm not trying to talk you out of the car. Indeed, maybe you can find other sights that are easier by car, to do while you have it. But rent the smallest car your family will fit in, because parking garage stalls and aisles are very narrow in Belgium (and most of Europe.)

I have driven in Leuven, and it's a challenge, with many narrow, one-way streets. Our hotel only had parking on weekends, when they had a deal with the neighboring hospital lot. Leven is a swell city to visit, with a beautiful Beguinage, a special town hall exterior, a nice art museum, and the (second ... ) replica reconstruction of the famous Catholic University Library. Costs are a bit lower in Leuven because of the large student population. It's a less historic city because of a huge fire 100 years or so ago.