Why is "Everybody" in the singular? We say "People in Europe are nice". Why, then, do we say "Everybody in Europe is nice"?
First, consider the sentence:
Every dog is nice.
This sentence is composed of a quantifier ('every'), a noun ('dog'), a copula verb ('is'), and an adjective ('nice').
Notice that because 'dog' is singular, the singular form of to be is used, 'is'.
Everybody is nice.
You can think of 'everybody' as being composed of a quantifier ('every') and a noun ('body'). It is a kind of compound quantifier phrase. Notice that the "embedded" word 'body' is singular. This should go some way toward explaining why one must use the singular of to be, 'is'.
The same pattern occurs with other compound quantifier phrases like 'everyone' and 'everything'. For example,
Everyone is nice.
Everything is fine.