Learn Finnish: Taikuri saa miehen nukahtamaan syvään hypnoosiin. - The magician gets the man to fall asleep into a deep hypnosis.

This is the core of the causative structure taught in this lesson. When you use saada to mean getting or causing someone to do something, the action they perform must be in the 3rd infinitive illative form, which ends in -maan or -mään.

Why is the object miehen in the genitive-accusative case instead of the partitive?

Because the action is completed successfully. The magician doesn't just try to get the man to sleep; he succeeds. In Finnish, this makes it a total object, which takes the genitive-looking accusative case (miehen) rather than the partitive (miestä).

Why do we say syvään hypnoosiin (into deep hypnosis) instead of just using the inessive case (in deep hypnosis)?

This is driven by the verb nukahtaa (to fall asleep). In Finnish, verbs expressing a transition or movement into a state often require the illative (into) case. You fall into sleep or into hypnosis in Finnish, rather than falling in it.

Could we use nukkumaan (from nukkua, to sleep) instead of nukahtamaan here?