I think you are confused with terminology.

1. You cannot make reservation through the rail planner app. Only via the Eurail reservation service (and not really recommended here) or other websites (mostly of national railway companies). These are completely seperate from the app, and are usually sent as a pdf.
2. You probably only added journeys to your pass, and activated travel days and generated qr-codes. This is not reserving, but registering your trains to a pass in order to have a valid ticket. Your pass, is the ticket. Please undo this, you never know if something unexpected happens, unabling you to travel that day. You should only activate these just before taking your first train of the day. Otherwise you lose these travel days, it’s not possible to de-activate a travel day on the same date. You can de-activate them, before reaching that date though.
3. Without details, we cannot guide you to the most adequate websites or places to reserve your trains. We can also give you tips on avoiding trains with mandatory reservations. Which trains are you talking about? When are you taking them? Between which stations?

An example of a trip I did last summer:

Circled in green is a train, added in “My Trip”, named Slovenië ‘22. I only got a train later to Brussels, than planned. This later train at 18:15, that I effectively took, was not initially planned. This is circled in red; So I looked it up in the planner while waiting for the train on the platform, added to my trip. After seeing the train rolling in at the platform, I tapped the grey ledger to activate my travel day and add the journey to my pass, and generate the QR-code for that travel day. We always advise this to do this, and not to add journeys to your pass, or activate travel days months before travelling.

I could board immediately, as this train from Ghent to Brussels didn’t require any seat reservation.

Different story for the train from Brussels to Vienna, also in the red circle. Note the “seat reservation required” message.

I booked this reservation, months in advance, since night train in summer are very popular. Via the ÖBB -nightjet website, the operator of the Nighttrain. But I only added the trip and selected the ledger just before travelling, to have a valid ticket together with my reservation for the bed compartment. The latter was a seperate pdf, that I printed out. So it was completely detached from the app.

Elsewhere, I’ve been advised to use Rail Europe to make my seat reservations for those trips requiring seat reservations in advance, as it is much smoother than Eurail Rail Planner. Opinions?

The Rail Europe website can't book any pass holder reservations.

Check these for advice on where to book reservations:

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

Here is my desired day 1 itinerary. all times are Central European Time. I found scheduled times at thetrainline.com - I realize train schedules may change somewhat between now and May:

20 May 2023

Copenhagen to Odense St., DB train depart 00:01 am, arrive 1:11 am

Odense St. to Hamburg, DB train IC 403 depart 2:35 am, arrive Fredericia St. 4:45 am

Fredericia St. to Flensburg, DB train IC 5751 depart 4:45 am, arrive Flensburg 6:07 am 

Flensburg to Hamburg Hbf, DB train RE 11159 depart 8:15 am, arrive Hamburg Hbf 8:45 am

Hamburg Hbf to Osnabruck Hbf, DB train ICE 107 depart 10:35am, arrive Osnabruck Hbf 11:14 am

Osnabruck Hbf to Hengelo, DB train RB 2-364 depart 12:26 pm, arrive Hengelo 12:26 pm

Hengelo to Utrecht Centrall, DB train IC 1746, depart 12:54 pm, arrive Utrecht Centraal 14:23 pm

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Am I able to take fewer trains and not pay more due to possessing a Eurail pass? Other things I should be concerned about? Some of these connections are tight.

Thanks,
Scott

Hi, I would not use thetrainline for planing. Use bahn.com in this case.

If you start later at the morning you will have direct trains between Copenhagen and Hamburg about every 2-4 hours. But after checking myself the timetable, there are direct trains missing on May 20th. Maybe there are works on the track.

For your desired route, and the options you give, I think you don't even need without seat reservations with those trains. As Angelo says, there are better options later on the day. 

The question is if its a comfortable option, with long waits in the middle of the night.

Can I use bahn.de to make seat reservations, as Seat61 recommends?

I’m trying to reserve a seat DB IC 433 directly on the bahn.de web site, and am getting the message “reservation impossible” … however I haven’t purchased a ticket, just trying to get seat reservation only, then use my Eurail pass as my ticket. 

You can book seat reservations on the DB website for the whole journey for just €4.50. But I'd travel a but later, honestly.

The time I wanted to travel - just to arrive sooner in Utrecht, to be honest - says “reservations impossible.” The later train, departing Copenhagen three hours later (3:01 am local time) does have an option for a seat reservation.

Your timetable doesn't match with what's actually running. E.g. there's no train Fredericia - Flensburg at 4:45 on 20 May.

If you ask the DB planner Copenhagen - Utrecht leaving at midnight, without any restrictions, then it will produce a timetable including the night train (Stockholm -) Odense - Hamburg, which it can't book online and that's why it says "Reservation impossible”.

But if you get DB to show a timetable including only trains with optional reservations, then you can book them all online for €4.50, e.g.:

So the night train doesn’t accept passengers at Copenhagen? Or I have to take my chances to board it without a reservation? Just trying to understand how that night train works.

Thank you for your insights!

Reservations are mandatory. It does not serve Copenhagen's main station, but the airport instead, leaving there at 00:05. But then it's best to stay on it until Hamburg. You can book a reservation on sj.se.

But first make up your mind about the trains you want to take. The timetable you posted does not exist.

I would add, that if you travel one day after there are direct day trains between Copenhagen and Hamburg. On 20 May they are missing.