Definitely, yes. I doubt many people in Russia would be familiar with the Farenheit scale :) In the UK I think most people can still use it, though I tend to use a mixture of imperial and metric measures. Anyway yes, this is Celsius.
Sure they would, e.g. on a weather report, for example. The "temperature" can be skipped in either language, but since it's there in the Russian sentence, it should also be present in the English translation.
I agree with Evelyn. Numerically it is written as -10, so pronounced "minus ten." In the States it's also OK to say "ten below" as shorthand for "ten below zero" but you would never say "ten minus zero degrees." The reason being that subtraction is non-commutative.