Ewww. Interesting, but weird to think of 10K year old blood. This guy, Semyon Grigoriev, is somewhat obsessed with Mammoths so I take his reports with a grain of salt but it would be difficult to mistranslate 'blood coming out'
"Ten below", "ten below zero", and "ten degrees below zero" all seem natural to me (BrE speaker), but I'm not sure about "ten degrees celsius below zero". That strikes me as an attempt to shoehorn units into an ambiguous* idiom. It doesn't work, and "-10C" or "minus ten celsius" would be better, in my opinion.
* Both F and C are used in the UK, but I think it's an age divide. I doubt many people under around 40 would use Fahrenheit as their measurement system of choice. I'm 35 and I have no intuitive understanding of Fahrenheit temperatures.
I'm British and read that without thinking anything of it, but I can see where you're coming from - it could mistakenly be read as "plus ten degrees below zero" perhaps.
I would argue that the unit of measure is "degrees Celcius" rather than individually degrees or Celcius. So I would agree with either ten-below-zero degrees Celcius (which is a little awkward), minus ten degrees C, or as the author wrote it.