As a EU citizen living outside the EU. Can I participate in elections for the European Parliament? Does it depend on my country of citizenship? My citizenship is Dutch.
Yes, it's possible but as you suspected it does depend on your country of origin. For example, French people abroad can simply vote in their consulates, the same way than for other national elections. Their votes will then be added to one of the districts (namely Paris/Île-de-France).
The modalities will depend on your citizenship (not all countries vote on the same day, some offer postal voting, some do not, etc.) and some countries don't even provide a way to vote for their citizens abroad but at least it's not forbidden as far as the EU is concerned. On the other hand, EU citizens living somewhere else in the EU can always vote in their country of residence.