Train from Brussels to Dusseldorf

There are a few things to consider here.

1. The standard fare is a “flexible” fare. You get a ticket valid for a certain day, and can get on any train on the booked route on that day. So if your train is late, you can wait for the next one. An exception is if you choose to book one of the “savings offers”, they frequently are only valid for the booked train (“Zugbindung”). But I think that you are still allowed to travel with the next train if you missed a connection by fault of the railway (your earlier train was late).
   

2. If you are worried about not getting a connection (because you booked a ticket with Zugbindung, or because there is no convenient next train) there is an option on the Deutsche Bahn booking page to specify the minimum number of minutes for a connection. The default will offer you as little as 5 minutes changing time, but you can change it and the system will only offer you the appropriate connections.
   

3. Connections from an ICE to a regional train are rarely problematic. ICEs are given priority for routing and don’t regularly run late, and if they do, it is by about 5 minutes. If they do run late, it is frequently due to some major problem, and then it can be really long delays, you cannot very well defend against those. Regional trains will often wait for a late ICE so people can make their connection if they hurry.

I cannot tell you if the flexible fare or the train-specific fare is better for you, consider both (including the price difference and your anxiety levels when traveling) and decide what is optimal for yourself. If you find it confusing picking the right fare on the website, you are likely to be able to book a ticket by phone. This will cost you more though, and you have to allow time for the ticket to arrive by snailmail.

If you have a ticket for a German regional train you can use it for all regional trains (not ICE or EC/IC) the same day. If you are in Aachen or Köln, there are many trains going to Köln or Düsseldorf, so if you miss one the next will come soon. And as others have said, no reservation on regional trains.

If you want the cheapest connection:

Look at the site of DB.
https://www.bahn.de/p/view/index.shtml
switch it to English (up in the first line left of the center is written Deutsch, clicking on it gives you a choice of languages.
Fill out start, destination and the date/time fields.
Below these lines untick the field prefer fast connections.
Go down to the grey field further options
on the new page opening go to Connections > more means of transport
Click on more means of transport.
Then some lines appear, untick the box ICE (this excludes all Thalys and ICE trains).
Then you get cheaper trains, Belgian IC and German regional trains, but with an absurd in-between connection between Welkenraedt (B) and Aachen (D).
This is the cheapest connection AFAIK. Takes 3:30 hours.
If this is too long for you, take ICE or Thalys.

You can book this, it will be fine.

For the ICE (Inter-City express) part a reservation is optional. If you want to be sure of a seat, make a reservation. If you don’t you may have to search a while for a seat that isn’t occupied, and if the train is very full you may end up standing or sitting on your luggage next to the doors.

For the RE (Regional Express, local train) there are no reservations, you just find any free seat, or stand, or sit on the steps if the train is very full.

If you miss your connection you can definitely take the next RE or RB to Düsseldorf. You may also be able to take an ICE between Köln and Düsseldorf, as you have already paid the extra ICE price for most of your journey. If you are not sure, show your ticket to one of the ticket-inspectors and ask before you get on.

The first train is a high speed train and requires booking. The second train however, seems to be a ‘normal’ train, which doesn’t require a reservation. You can see this in the train service number, high speed trains are ice, thalys, eurostar, izy and tgv. High speed trains require reservation in most cases, except national ICE trains.

The re____ train service number stands for regional express. These trains are just national rail and don’t require a reservation. (As @DCTLib noticed in the comments) Another option would be to book the 2 trains independently. This way you could still have the reservation, but choose how much time you leave between the trains.

Sources:
German railways website, train travel in Belgium, The Netherlands, which is similar to train travel in Germany