If you have to test or not changes weekly bestway ist to check the Website of reopen.eu or even better the individual Foreign and/or Tourist Ministery Websites of the countries.

What about the new EU-wide Covid status registry?  Are non-EU travelers entered in the registry when they first arrive in the EU?

Don't know if I got the meaning right, but there is no general registry for all of europe, that every county can make searches in. You can get an QR Code and with some apps you can create your own digital vaccination pass. Or a paper one that you get from your doctor or pharmacys. With them you can vaildate that you are vaccinated(believe you have to show this in every country). Don't know how a citizen from the US could get it...maybe you can check what your health ministry is saying on there official website or some from the EU?

The website is actually reopen.europa.eu; it is a good starting point, with further links to the relevant national websites.

I'm not sure in how far tourist travel to the EU is already allowed again and in any case, every country has its own requirements because health care is a national competency.

Some allow it, but as you now can read in news papers, the delta variation is spreading and for example german tourists are now heading back home, because of the situation in Portugal. Norway is still closed for tourists, the UK have high numbers. It's best to check the national websites of the countrys and decide then what to do. 

Polkadot:

Re "new EU Covid status registry"

https://ec.europa.eu/info/live-work-travel-eu/coronavirus-response/safe-covid-19-vaccines-europeans/eu-digital-covid-certificate_en

Each certificate issued is registered in an EU--wide registry.  All EU members will be online and issuing certificates by July 1st.

Nowhere in the article does it state that EU Digital Covid (Green) Certificates are only issued to EU citizens.  Nor does it explain how or if non-EU travelers can interface with the new system.

NEW News:

https://www.forbes.com/sites/suzannerowankelleher/2021/06/03/the-eus-digital-covid-certificate-could-soon-be-available-to-americans/

New EUDCC news:

https://www.stripes.com/theaters/europe/2021-06-22/digital-vaccine-certificate-available-to-americans-in-germany-1767374.html

So U.S. travelers can get EU Covid travel certificates at EU Embassy and counsulates

Question, I am planning on using the rail from Copenhagen to Berlin, and after 3 days head to Prague and after 3 days across to Vienna and after 5 days head back the same route in one straight shot of 21hrs. With a PCR validity of 72hrs, do the different locations have testing centers for non-residents in transit?. 

You failed to specify your nationality or your Covid vaccination status.

International travelers now traveling internationally and within the EU must now must rely on the new EU Digital Covid Certificate gateway.

If you are a vaccineed.U.S. traveler then EU countries will honor your CDC vacconation card and (hopefully) add you to their EUDCC EU Covid status database.

There are currently NO trains from Norway to Denmark via Sweden.

Norway will NOT welcome you with open arms this summer. 

The good news is that if you upset the authorities (easily done at the moment) the quarantine is only approx 75USD per night which by Norwegian standards is cheap for a 4 star hotel. The UK is £220 per night! 

“People who are fully vaccinated or who have recovered from COVID-19 in the past 6 months, and who can document this with a verifiable COVID-19 certificate that is connected to the EUDCC gateway, will be able to enter Norway freely, regardless of which country they are travelling from.

These people are exempt from the duty to quarantine, testing prior to arrival, testing at the border, and the requirement of entry registration.”

Current rules: https://www.regjeringen.no/en/topics/koronavirus-covid-19/travel-to-norway/id2791503/

There are currently no passenger trains across the Norwegian border, but buses and ferries are operating.

Good additional information. Thanks