Yes that is true that Romans took slaves from conquered countries. There were also other ways for someone in Roman society to end up in slavery, for example sometimes poor people sold their kids into slavery.

The ethnicity of slaves used as gladiators could have been the whole range of ethnicities the Romans came into contact with. There would likely have been many from North Africa, the Middle East, and virtually all parts of Southern, Central, or Western Europe. There might have been smaller numbers of people from further afield places like sub-Saharan Africa, South Asia, or Northeastern Europe.