As already pointed out in multiple comments this is very country specific and based on your description I cannot tell to which country do you travel. I can only rule out few.
Yet since you have received your (printed) tickets it means you are eligible to take a trip as long as you go according to the validity dates and destinations on the ticket. Since your tickets do not have specific time schedule, just validity dates and destinations most probably you are eligible to take any train in that range and most probably there is no seat reservation available for the trains you have selected. It might be that there is a seat reservation but it is optional and paid additionally.
While very unlikely, sometimes the rules for tickets sold abroad are different than tickets sold locally and your situation might be that you do not get a specific seat reservation despite regular country tickets have them. In such case you get trip flexibility at the cost that there might be no seat available for you. Usually it will mean you'll have to stand part of entire trip if no free seats are available. There are some trains (like EIP in Poland and possibly some other high speed trains across Europe) where you will not be allowed without seat reservation, but I expect for those trains you would get ticket with seat assignment included.
So in general, as long as you have tickets and their validity meets your needs, you're good to go with trains of your choice, possibly with some additional flexibility.
You mentioned your schedule is quite tight so your other concern might be delays. Here the situation is a bit trickier. There is an EU law that gives a passenger a number of rights (all listed here). Some of them include:
- be transported to your final destination at the earliest opportunity (or a later date of your choosing). This includes alternative transport when the train is blocked and the service is suspended.
- meals and refreshments (proportionate to the waiting time)
- accommodation – if you have to stay overnight.
Specific local implementation vary in EU member countries, for example in Poland law enforces train operators to ensure that in case of a delay you can continue your travel on the same day. So if your next connection is the last day to the destination on that day this next connection has to wait regardless how long the delay is. If the next connection isn't the last train to the destination on that day it's on railroad company discretion to decide what to do (in some countries it might vary and rules might be more strict for the railroad enforcing it to delay next connection anyway). It might be different in other countries and, as pointed out by Willeke sometimes you just get money for a taxi to your destination.
Anyway in such case it is good to check with the train crew if the next train can also be delayed allowing you to change.
I don't know about other countries but for instance in Poland you are entitled to rebook your ticket for a different schedule (within some timeframe, but it's something like 30 days if I recall correctly) at no cost.
Also since you bought all your tickets from a single travel agent (webpage) and as I understand in one transaction these rights for package travel also apply to you (unless all tickets are on the same day):
The organiser is responsible for the proper performance of all travel services included in your package.
If a travel service can't be provided as agreed, for example if a provider can't carry out an agreed service or can't do so in the agreed form (such as providing transport to or from your destination, providing the agreed type of accommodation, or carrying out a guided tour that you booked), the organiser has to resolve the problem at no extra cost to you.
If it is impossible to make alternative arrangements or you reject the arrangements offered to you on valid grounds, and the package includes your transport (such as air travel), the organiser must offer to repatriate you. If the travel services do not reach the agreed standards and this cannot be resolved on the spot, you may also be entitled to compensation.
You are also protected if for whatever reasons you cannot perform your trip due to the incorrectly issued tickets (on the same page):
When you book a holiday, the responsible trader (the travel agent or online travel agency) is liable if any of the following occurs during the booking process:
- errors made by the retailer if they are responsible for arranging the booking of a package or of travel services which are part of linked travel arrangements
The page does not state details about the liability.
Note that those last two excerpts state the liability of the ticket retailer, so the webpage where you booked your tickets. As you can see you're double-protected, at least that someone will either have to take you to your final destination (either train company or ticket retailer). Note that the liability (especially in case of tight schedule) is limited to reaching destination only (so it might be you'll be rerouted and miss some of the intermediary targets) and there might be rules limiting the liability if you left yourself inadequate time for a change.
Please note - all those rules apply in EU. Not all European countries are members of EU! Yet majority is and you can easily check that. Also country might decide not to apply those rules for local connections but since you're referring to IC tickets that's probably also not the case.
Edit based on the specific trip leg provided by OP and some extra research
I couldn't find details about operator that operates the Brussels - Luxembourg trip but I've checked conditions on various sites (SNCF, SNCB, RailEurope). Each of those pages gives you specific details about fare for each train. For this specific train the fare is open, meaning you can choose any train within the given range of dates. Also on this route there is no seat reservation offered, which means in general you're free to choose any seat available (within your class of course) and once seated you cannot be requested to leave your seat by someone claiming to have seat reservation (unless you pick a special seat that in general is reserved for pregnant women, mother with children or people with some disabilities - check the stickers over the seats that informs about that and I suggest - avoid those seats). Moreover Rail Europe claims the ticket is valid for 15 days, giving you a huuuge flexibility:
From BRUXELLES NORD BR To LUXEMBOURG LUX
Eurocities 2106 - Reservation not included
1 x Ind. Open Ticket Adult
European open ticket must be used within the 15 days validity period. Journey might be effected on any date within the period of validity. Open Tickets do not guarantee a seat, thus reservations are highly recommended. Important: Reservations are mandatory on most of high speed, scenic trains and all night trains.
The scripted highlight is mine.
Also the same train checked on SNCB page (that is IMO most accurate for this particular leg - I believe this is the actual operator of this specific train, but it's just a best guess based on my knowledge of regional transport) shows those conditions (to obtain it simply use the trip planner and follow with the ticket purchase up until you have a specific train selected):
Bruxelles-Midi/Brussel-Zuid - Luxembourg € 43.60
Adult (26-59): 1 € 43.60 per person
Flex Fare Adult 2nd Class
Special information: Ticket valid for any conventional train on the requested route during the whole validity period of the ticket (for homeprinted tickets, the validity period is limited to the requested travel date). Not valid on high-speed trains, except for ICE trains.
Refund Conditions: Refund possible for tickets presented up to the first day of validity. After the first day of validity, non refundable.
Exchange Conditions: No exchange possible
Travel Conditions: no specific travel conditions. ATTENTION: homeprint tickets have a validity of 1 day!
Again, scripted highlight is mine.
On this SNCB page about passenger rights you can read more about carrier specific implementation of those rights described earlier. I'll not quote it here, as this will lead you to the most up-to-date terms and in case of any changes in the future the quote might become misleading.
Finally this nice SNCB page about terms and conditions provides links to many carriers T&C details that should provide final, definite information if you're still in doubt. I just need to warn you, that it's sometimes not so obvious to find out which specific fare conditions apply and some of the pages are available in the carrier language only.
One last remark - it is often possible to buy a ticked without seat reservation for a train with either optional or mandatory seat reservation. In most cases (from my experience all trains other than high-speed and some express class, definitely all the standard ones that will be marked just with IC in the train number) such ticket is enough to have a trip and usually it gives you a possibility to obtain a seat reservation (for free if it is compulsory for a one-trip tickets bought directly at the carrier, paid if the reservation is optional or mandatory but paid additionally - not sure if this option is still used anywhere but yes, there was something like that in the past). The channel to obtain such seat reservation depends on your ticket and a carrier, in general you should be good to get it any any counter selling the tickets of that carrier, and for sure in the country of the operator.
If the seat reservation is not possible with a regular fare for that route you'll not be able to get it in any way for other than standard tickets as well.
If for some specific leg you're still unsure or can't find the details as I show here, try calling the carrier in the country of train departure. They will for sure be able to give you most accurate details given you provide the train number you plan to go with.