I am a EU national and UK resident. I need dental care but no GP care. I understand that a GP can provide me with an NHS number. Do I need to register with a GP in order to get an NHS number in order to get NHS-priced dental care? Or is this not needed, and can the dentist provide me with an NHS number — or do I not need an NHS number at all for dental care?
To get any kind of NHS treating the easiest way is to get a GP. You should be able to find one in your area quickly, and registration takes around 30 minutes. Not having a GP means that even if you are entitled to NHS care, you can get into problems. They usually ask for your NI number (which is different to your NHS number) when registering with a GP, and once you do, you will get your NHS number a bit later by post.
A dentist is not a GP. Actually, when I first registered at my dentist, they handed me a form, and one of the questions on the form was the address of my GP. They didn't need my NHS number nor my NI number (and they usually don't expect you to remember those).
With this in mind, you should definitely get a GP in your area. Once you do you can just go to the dentist and register there as well. The name and address of the GP should be enough. You should already have an NI number though.