All of those trains are unreserved and you are free to use with a pass.

Why are they all greyed out? It was like that from Germany to Amsterdam and the train was crowded but it worked out. However, I asked a girl in the DB Reizecentrum in Hamburg and she said that it was harder to go without reservations in france, and I’ve heard the same thing from a friend. I don’t wanna miss my booked train to Nice so therefore I want to be sure.

All this trains is not possible to reserve a seat. I am not sure for the TER in France, because every Territory in France has other rules for regional trains like TER. 

The str8forward route from AMS to Lux is daft easy: only 1 change from hourly train to same: at Brussel Noord-unless you have specified maybe a special scenic route? or maybe just on that day there are works or whatever going on. OR via Maastricht-LUik (LIege)=change there, but connections not very good).

BTW; IF you proceed from LUx (where all transit is free) there is no need at all to go via Paris to go south-TGV toward Marseille-Nice also start from Strasbourg (much less though) . The se planners however often have a tendency to send people via expensive extra payment routes.

The ONLY routes were a RES on TER is-sometimes-needed are the Nomad trains from Paris to Normandie=far away from this routing.

Note however that PACA region (around Marseille-Nice) has STOPped accepting passes since 1/7!!!