Learn Russian: У этих опасных собак строгий хозяин. - These dangerous dogs have a strict owner.

This sentence uses the standard Russian way to say "to have": У + [owner in genitive] + [object owned in nominative]. Literally, it translates to "At these dangerous dogs [there is] a strict owner."

Why does этих end in -их but опасных ends in -ых if they are both genitive plural?

The demonstrative pronoun этот (this/these) takes the soft genitive plural ending -их, making it этих. Most regular adjectives with a hard stem (the consonant before the ending), like опасный (dangerous), take the hard genitive plural ending -ых, becoming опасных.

I see the adjective and pronoun have plural endings, but why doesn't the noun собак have a visible genitive plural ending?

Most feminine nouns that end in -а in the dictionary form, like собака (dog), drop that -а to form the genitive plural. This leaves the word with what we call a "zero ending," making it just собак.

Why is строгий хозяин in the nominative case? Shouldn't it be the direct object (accusative case) since the dogs "have" him?

In English, you use the verb "have" and a direct object. But in the Russian У construction, the person or thing being "had" is actually the grammatical subject of the sentence. Because of this, строгий хозяин (strict owner) must be in the nominative case.