Learn Russian: Когда мне некогда отдыхать, мысли вылетают из головы. - When I have no time to rest, thoughts fly out of my head.

Using вылетают (fly out) is a common metaphor for losing your train of thought or forgetting something. It is imperfective here because it describes a recurring event. Remember that for prefixed verbs of motion, adding a prefix to the multidirectional base летать creates the imperfective (вылетать), while adding it to the unidirectional base лететь creates the perfective (вылететь).

Is некогда just a typo for никогда (never)?

No, they are different words! никогда means "never" and is used alongside a negative verb. некогда, with the stress on the first syllable, means "there is no time". It acts as a standalone predicate and requires the dative case to show who is experiencing this, which is why we use мне (literally: "to me there is no time").

Why is отдыхать (to rest) in the imperfective aspect here?

After некогда, we usually use the imperfective infinitive when talking about a general, ongoing activity that we lack time for, like resting. You would generally only use the perfective after некогда if you had no time to accomplish one single, specific result in a specific moment.

The English translation says "out of my head", but the Russian sentence doesn't include a word for "my". Why?