Is slaughtering a sheep by placing the knife on the middle of the neck, while uttering the name of Allah and glorifying Him, and placing the right foot on the neck, the correct way of slaughtering?

The steps taken by your brother are correct and lawful, as he combined the conditions and etiquettes (of slaughter). As long as the knife severed the jugular veins (trachea, esophagus, and the two surrounding blood vessels), there is no reason for doubt.

The Messenger of Allah, peace and blessings be upon him, sacrificed two horned, white-black rams, slaughtering them with his own hand, saying "Bismillah" and "Allahu Akbar," and placing his foot on their sides.

One etiquette remained unmentioned: facing the Qibla with oneself and the sacrificial animal.