I don´t think it´s the same throughout Europe. For Germany, and if you don´t earn more than € X per month, the relevant date is when you get your earnings, which would be the moment you have it in your PayPal account (or even when it´s cleared by Fiverr or even the “billing date” may be relevant if you are over the limits in Germany).
The German Finanzministerium publishes a monthly conversion rate for all currencies (IIRC it´s based on the European Central Bank’s rate, so that might be similar in other European countries), which one can use for convenience, so you only need to do the maths once a month for the amount earned by month, but in theory, I could use the “official” conversion rates for all the dates I received money in my PayPal account.
I think it would be best to find someone who is a pro in tax things (and ask them if they have experience with “online freelancing”, earnings in foreign currencies) in your country and pay them at least once to answer all questions/tell you what you need to know, so you´re on the safe side and can do what you have to on your own from then on.
You may even have to let a pro do your yearly tax declaration at least once (even if it is super easy to do yourself) if your health insurance demands that for figuring out your dues.