I'll preface this with my understanding of particles that I think are relevant in this context.
「を」 - point of departure, or movement through a location.
「に」 - specific point in time, or direction of motion.
「で 」- point of action.
For the motion related verbs (e.g. run, walk). It's seems straightforward.
「雨の中を歩く」"Walk in (through) the rain."
For the verb 「出かける」. Normally it is used with 「に」 to indicate where a person leaves to. e.g. 「買い物に出かける。」
However, the only instances of 出かける being used with 「を」 instead of the usual に is in the phrase: 雨の中を出かける
Which usage of 「を」 is this? It certainly does not mark the point of departure. Maybe it's the second usage? You leave and "go through" the rain? I'm not too sure.
Similarly, 「を」 is not being used to mark the point of departure in this sentence.
Lastly, the sentences that confused me the most.
Not only are none of these motion verbs, all 3 particles seem interchangeable? In particular the last one, which uses both 「を」 and 「に」 . I'm very confused as to how these particles were chosen.
Evidently I seem to be ignorant of some grammar that would explain all this. Would very much appreciate an explanation. Thanks!