Every one that believeth (panta ton pisteuonta). This accusative active participle of general reference with the infinitive in indirect discourse is the usual idiom. Only labein (second aorist active infinitive of lambanō) is not indirect statement so much as indirect command or arrangement. The prophets bear witness to Jesus Christ to this effect. It is God’s plan and no race distinctions are drawn. Peter had already said the same thing at Pentecost (Act 2:38), but now he sees himself that Gentiles do not have to become Jews, but have only to believe in Jesus as Messiah and Judge as foretold by the prophets. It was glorious news to Cornelius and his group.
Through his name (dia tou onomatos autou), not as a title or magic formula (Act 18:13), but the power of Christ himself represented by his name.