Eurail Reservations

Reservations are IME available via Eurail sometimes, but not always. In England and Scotland, we needed to reserve seats directly with the railroad company we used for the trips. France and Spain I was able to get seat reservations through Eurail, but they would have been cheaper in some cases had I secured them directly with the carrier. When reserving seats with the carrier, in most cases they required the transaction be in person at a station.

Eurail does not always allow you make seat reservations, buy seat reservations from the actual companies running the trains instead. But if those four trips are all that you are planning, I'm not sure a pass is good value.

For these trains just make reservations at the station once you are there. If you really need to nail those trips down months in advance you should not have gotten a pass in the first place. Note that July is so far in the future that you wouldn't even be able to get tickets, let alone reservations now.

So Bergen to Oslo: Just get the reservation at the station in Bergen.
Oslo to Copenhagen: No need for reservations (which are probably not even possible on that route)
Copenhagen to Amsterdam: No reservations needed, but you can get them if you want in Copenhagen.

Do not listen to the advice above, getting reservations at the station means that there is a risk that the trains might be sold out and you have to wait for one that has free seats.

Also, reservations are mandatory from Copenhagen to Amsterdam, so don't try to board the train without one. And from Oslo to Copenhagen, there are two options from Gothenburg to Copenhagen, regional trains with no seat reservations and high speed trains with mandatory seat reservation.

My only Eurail experience was for the regions you mentioned, and my experiences are exactly as Badger states. Making reservations were mandatory and not included in the Eurail cost. I made mine the day prior to our Oslo to Bergen trip and we had different seats for different portions of the journey. This was July 2016 - and things do change.

Trains at least occasionally sell out between Oslo and Myrdal, which is on the Bergen line. I'd be concerned that the eastern part of the Bergen-Oslo trip might also sometimes sell out.

I have travelled that region on an Interrail pass myself. And I booked all trains a few days beforehand at the station. if trains did sell out regularly days beforehand they would be useless.

We spend a few days in Bergen, and did the a circuit Berngen - (by boat to) Flam - Myrdal - Bergen. We just booked that in Bergen, and use a pass day for the Myrdal - Bergen part. Bergen - Oslo and Oslo Stockholm we booked at the same time.

You do not need to be in Bergen btw. to book the Bergen - Oslo train, you can do that at most stations, so you can book a few legs out. In many cases we would book our next leg when we arrived, as arriving puts us at the station anyway.

Note that there are no high speed trains from Gothenburg to Copenhagen. (There are a handful to Malmo though). On Oslo - Copenhagen you will be on a regional train till Gothenburg, and on a Oresundstag commuter train to Copenhagen from there.

It is however absolutely ridiculous to now already worry about trains in July. In many cases the railways don't even know what train they will run. And if you really are the kind that nails things down months in advance you shouldn't get a pass in the first place.

if trains did sell out regularly days beforehand they would be
useless.

Call them useless if you like, but that does happen. Especially Bergen-Oslo and Copenhagen-Hamburg are very popular in the summer and regularly sell out.

Note that there are no high speed trains from Gothenburg to
Copenhagen. (There are a handful to Malmo though).

Note that there are (probably). SJ runs high speed trains from Gothenburg to Malmö, and in the summer they are usually extended to Copenhagen.