How to install Lightroom 5 purchased on CD in 2013 on new PC?

That is what I would do - buy an external CD ROM drive! OR go with the Subscription and keep up to date! Lightroom 5 worked on my Win 10 platform, but now use the CC version (esp as I have a more powerful computer).

You couldn't do panorama shots on Lightroom 5 among other useful functions, but with the later versions, you can. This is a really useful function!

Purchase an external CD ROM drive, these can be found for around $25 US. Or perhaps you can find the download file via these instructions: https://helpx.adobe.com/download-install/kb/downloaded-older-app.html

... I was running LR 5 on my old computer for many years but had to buy a new laptop recently to move to Win 10.

The new laptop - of course - has no CD-ROM drive any more. Since I cannot use the installer CD I bought in 2013, which I still have including serial number, I was looking for an installer to download from Adobe's website.

Adobe has removed the installation files for all older version from their public web sites. If you bought your Lightroom version from Adobe you canc find the installations file under your Adobe account.

If you bought from a third parrty reseller you have to contact Adobe directly for a download link. Have your serial number and details of original purchase available. Make it clear that you doesn't need support for an older product but need a download link. Click the link below and scroll down the page to Individual “Sign in to contact us” It should be possible to start an on-line chat with an Adobe agent.

However, please note that Lightroom 5 isn't compatible with the new Windows 10. It's possible the LR is working fine, but it isn't guaranteed.  

But it seems that my version is so old, that it is no longer provided / supported.

Yes you're right. There's no more bugfixes and updates to support new cameras, new operating system and so on.

So, I installed the Creative Cloud and installed LR Classic from there in the hope that I can activate the app with my original serial number. My LR5 version is registered with Adobe and associated with my Adobe ID, I can see that in my account. But that does not seem to work either, I can only test or buy. 

You can't install and activate Lightroom 5 via the Creative Cloud Desktop. There's no way to get it work and thwere's also no way to get an update to the CC version. You have to buy a subscription.  

As I am perfectly fine with my LR 5, I do not feel the urge to upgrade. The only workaround I can think of right now, is to buy an external CD-ROM drive, plug it into my USB slot and use the old installation CD. Would that work?

That can be worked, but as I said above LR 5 isn't compatible with Windows 10. 

In order to be on the safe side and to be able to work without problems in the future, you should consider switching to the CC version.