Learn Russian: Дети везде. - Children are everywhere.

In Russian, the present tense of the verb to be is almost always left out. Instead of saying 'Children are everywhere,' you simply say Дети везде (Children everywhere).

I know дети means 'children,' but what is the singular word for 'child'?

The singular is ребёнок. This is one of the most irregular plurals in Russian, where the singular and plural forms look like completely different words!

Could I change the word order and say Везде дети?

Yes, you can! Russian word order is flexible, but the word at the end usually carries the most emphasis. Дети везде emphasizes where the children are (The children are everywhere). Везде дети emphasizes who is everywhere (Everywhere, there are children).

Are there any other common words for 'everywhere' besides везде?

Yes, you might also see всюду. They mean the exact same thing and can be used interchangeably, though везде is slightly more common in everyday conversation.