Eurostar trains Brussels - Koln require reservations, DB ICE and other connections do not.

You didn't mention a date (which you should because it depends on it). 

From 1.6. to 1.9. a reservation is mandatory for ICE 15. 

Rest of the year: no reservation required.

Thanks for the advice, it is 13.7.24 so mandatory. Does that apply also to other high speed trains, thinking of Koblenz/Salzburg, Salzburg/Chur, Feldkirch/Vienna, Vienna/Prague, Prague/Munich, Munich/Brussels

It does apply to many long distance trains (not: regional trains) but you need to have a look for each one.

For the trips mentioned: 

It’s possible to go to Salzburg Hbf itself from Germany without a mandatory reservation, but a reservation is needed if you go from Germany beyond Salzburg Hbf.

No mandatory reservation is needed from Salzburg Hbf itself to Chur.

No mandatory reservation from Feldkirch to Wien.

No mandatory reservation from Praha to München.

Mandatory reservation from München to Bruxelles for the train crossing the border from Germany into Belgium.

For all trips: That does just apply to long distance trains, not regional trains. Regional trains are without reservation.

The Brussels to Koblenz is ICE15 to Koln then RE28515 TO Koblenz. It is not Eurostar, so does it need a reservation?

The Eurostar train does always need a reservation (all year long).

As written before: ICE 15 has mandatory reservations 1.6. to 1.9.

Thanks Hektor for the advice 

Also want to add that even if those trains do not require reservations I'd make them to be sure. They are very popular (and often tiny...) so you might have to stand otherwise.

In the worst case staff might ask people without reservations to leave the train due to overcrowding (in some occasions they even have to call the police...). Of course it's rare but it definitely happens every summer.

Please note that Feldkirch - Buchs SG (towards Chur) will be closed for 3 weeks in August for works. Rail replacement buses will run, expect longer travel times.

Thanks everyone