Overnight trains from Paris to Copenhagen.

This might help you. 

https://community.eurail.com/travelling-by-train-65/paris-to-copenhagen-night-train-5983

The night trains that come to mind are Basel-Hamburg (also stops in Offenburg and Karlsruhe, although KA is already after midnight) and Hamburg-Copenhagen/Malmö(-Stockholm).

Basel-Hamburg does not have couchettes or sleepers anymore on the 17th. Seats are still available.

The SJ night train Hamburg-Malmö also only has seats left. But it's in Malmö already at 3:52 so that's quite early. Or at Copenhagen airport even earlier…

The Snälltåget night train from Hamburg still has couchettes available. Book ASAP here:

https://www.snalltaget.se/en

It stops in Ørestad at 6:47, from where it's a short ride with train or metro to Copenhagen.

I have a 2 month Global Pass and would not mind taking a detour via Basel as suggested but would like to avoid transfers in the middle of the night. 

So I found a 20:59 from Zurich (NJ 470) but it is showing First Class N/A but on the "Show Facilities" screen it does show First Class options as shown in the screenshot below... what's is to be made of this discrepancy? Would like to book 1st class if possible with the First Class Global Eurail Pass.

As written above the night train Zurich - Basel - Hamburg is full on 17th April. Nightjets can sell out weeks in advance. Interrail isn’t the best place to look because it doesn’t show availability.

You could spend the night on seats but it is really not recommended. The seats aren’t comfortable for sleeping and lights will be fully on.

An itinerary :

• Paris-Est - Strasbourg 07:58 - 09:59 10€ seat reservation (1st class)
• Strasbourg - Offenburg 10:51 - 11:22
• Offenburg - Hamburg 11:27 - 17:36
• Hamburg - Copenhagen (or Stockholm) 22:43 - 06:47 (Oerestad near Copenhagen) 174€ (2000 SEK) for a private couchette compartment. Book ASAP on snalltaget.se as advised

Many thanks @thibcabe for the clarification.... your sample itinerary sounds like the best option available for me at this point. 

Would like to understand the discrepancy between 1st class N/A but still available under the "Show Facilities" option... I have noticed such discrepancies elsewhere too.. for example this night train from Paris to Nice below: 

N/A to me would mean to me that there are never 1st class seats on that train.

Btw there would be another option for you. More expensive but you leave Paris later :

• TGV Paris-Est - Strasbourg 09:25 - 11:11 20€ seat reservation (1st class)
• RB Strasbourg - Offenburg 12:50 - 13:22
• ICE Offenburg - Hamburg Hbf 13:28 - 19:36 free 1st class reservations on tickets.oebb.at (good idea if you want to sit together but not mandatory)

There are at least 2 later trains to Hamburg if the RB is delayed (shouldn’t happen)

No, N/A just means that the app doesn't have a price. It may mean it doesn't exist in this train but in this train there is a first class carriage, running as part of IC 60470. Price of a seat is also wrong, as is often the case in the app. I usually ignore any price mentioned there.

I just registered at tickets.oebb.at and was wondering if an Austrian website would be better to make reservations between Switzerland and Gernany routes? 

Also, can someone point to an exhaustive ist of all such websites (like tickets.oebb.at) that are recommended over the Eurail sites when confirming reservations?

Regards.

tickets.oebb.at is a good source.

Check this thread:

• How to get Reservations :)

and this page:

https://www.seat61.com/interrail-and-eurail-reservations.htm