You can use any of the train companies included in the Eurail offer.

I see that you already have found the excellent guide to how to make reservations. 

The experienced travellers in the group have made a list with some advice that hopefully will be useful. 

Here is some useful information from the experienced travellers in the Community regarding both planning, reservations and activation of pass and travel days. 

 Planning

The rail planner is normally not up to date, as it only is updated once a month, so to be sure of the time table you better check the timetable and availability on the websites of the national railways. The bigger national railways, like DB (Germany) SBB (Switzerland) and ÖBB (Austria) cover several countries. 

 Reservations 

The map is just to get an idea :)

You can ofcourse take a more direct Route like Amsterdam - Osnabrück/Münster - Hamburg Or Amsterdam - Hannover - Hamburg and then onwards to Copenhagen

I recommend to use Railplanner Apps (Bestwise the local app but atleast a major app like DB Navigator (bahn.de), Scotty(oebb.at), SBB Mobile(sbb.ch ), Muj Vlak (CD.cz) and just use the Railplanner of Eurail as traveldiary and to inform about reservations needed or not.

Thanks @seewulf. 

So what is the best way to find out this kind of information - for example, I would not have known about going from Amsterdam → Osnabruck → Hamburg. 

And also to do this trip - how do I know which rail company I can use with my Eurail pass and which rail company I should use?

Thanks

If you check German railways bahn.com you get often in the planner the train company that is running the train and also if you need a reservation (reservation compulsory or global price train)

DO NOT GIVE THIS COMPANY YOUR MONEY.

I find it useful to use a few tools to develop my travel routings and this may help:

All rail planners (Interrail, DB. OEBB, Trainline etc) use algorithms for complex journeys including time from A to B, Number of changes, Transfer times etc and usually only show the routings and trains that get the highest scores, discarding many combinations and routings at similar times.

I look at rail maps for obvious alternative towns and cities on my way from A to B and then check for example A to C, then C to B, maybe for a stopover or even a night there.

For all the suggestions for A to B I also expand to see all the stops, not least of which to see the route it follows. In Germany for example there tends to be regular ICE trains A to B but they take different routes, offering both choices and the ability to avoid the slower routes.

This sort of approach reveals some great options.

On a recent journey this approach showed that Siracusa (Sicily) had a direct train to Rome that took 10 hours and arrived well into the evening (about 2030) and we wanted to go to Bologna the next morning. Checking stations along the route we identified Salerno as arriving early evening. Then checking Salerno to Bologna we found that Salerno was the first station in the high speed line across Italy and had direct links to Bologna. Rome wasn’t high on our list on this journey, merely the stopover recommended by the planner.

Later on the same trip we arrived at Munich to transfer to Nuremberg. Our train had a fault and we were advised there was another ICE leaving about 10 mins later to Nuremberg. This had not shown on any planner but we hopped on, only to realise this had a totally different routing that took nearly 2 hours v the first train routing scheduled for 1 hour 5 mins. (OOI our original train actually arrived at the same time as the one we took so obviously took nearly an hour to solve the fault).