Once Hauken's Insight leaves the battlefield, the player that controlled it may no longer play the exiled cards. If it's destroyed and somehow returns to the battlefield, it's a new object with no memory of the previously exiled cards.

Paying {4}{U}{U} and transforming Jacob Hauken, Inspector is part of the resolution of the activated ability. You draw a card and exile a card before choosing whether to pay.

Playing a card using the last ability of Hauken's Insight follows all the normal timing rules for that card. For example, if you play a land this way, you may do so only during your main phase while the stack is empty and only if you haven't yet played a land (unless another effect allows you to play additional lands).

You may play cards this way that were exiled with Jacob Hauken, Inspector before it transformed into Hauken's Insight.