Is it wrong to have "he ovat"?
He means just they. The subject is the child and the dog here.
Can you drop "ovat", or is it mandatory?
"ovat" is mandatory. The only thing that you absolutely must have to make a sentence in Finnish is the predicative verb, in this case it is the "ovat"
It is the verb of the sentence so it is necessary. Otherwise it would mean The child and the dog - always together.
In Russian, it would not only be unnecessary, it would even be wrong to say it.
Unless you're placing emphasis. But yeah, normally the verb 'to be' is not used in these situations.