grammar - Function of participle

Those are some very good questions. They are not always entirely clear, partly because you don't always define your terms, such as "participle adjective"; but I will answer as best I can. The definition of "clause" that I use is the traditional one: a finite verb and its dependencies. By that definition, a participle and its dependencies don't make up a clause, but a phrase, hence "participial phrase". I am aware that some people use different definitions, and each definition has its benefits; but this is the one I have chosen.

Ad 1: yes, both are participles, and all participles function as adjectives to some degree, in that they modify a noun (or replace one, just as normal adjectives can when used substantively). Externally, participles function like adjectives, since they depend on nouns; internally, they function like verbs, because they can have verbal arguments that depend on them, just as other verbs have them (like objects and adverbs).

Ad 2: yes, the present participle generally describes something that happens at the same time as the main verb (but it's complicated).

Ad 3: a present participle can be equivalent to either continuous or simple clauses. I'm not a fan of the term "reduce" in linguistics, because it implies an actual historical or mental process of reduction (from clause to participle) that I believe never happened. By equivalence I mean two constructions that are different in form but have the same meaning (different syntax but same semantics).

Ad 4: again, I object to this notion of "reduction". In my opinion, there is only equivalence, and equivalence never has the last word in syntactic analysis, because you can often create several, different constructions equivalent to whatever phrase you're analysing.

What those books mean is that participles can be replaced by equivalent relative clauses, but also by equivalent adverbial clauses, as in your example, and also by other constructions. This is not unique to participial phrases.

The lying man deceived us. (participle)

Lying, the man deceived us. (same as above)

The man who lied deceived us. (relative clause)

The man deceived us when he lied. (conjunction: adverbial clause)

The man deceived us with his lies. (simple adverbial phrase containing a noun)

The man deceived us by lying. (simple adverbial phrase containing a gerund (gerunds function like nouns))

In your example, passing is simply a present participle modifying the subject, he. The only unusual aspect to the example is that the present participle can be said to happen before the main clause, which is normally not possible with present participles; but that is debatable. At any rate, that aspect is not relevant to adverbiality in general.

Ad 5: that depends on your definition of "tense", which is a hot topic in linguistics; but I would say that participles don't have tense according to most definitions. They don't refer to a specific time: the time to which they refer generally depends on the time to which the main verb of the sentence refers. A present participle refer to roughly the same time as that of the main verb, a past participle to a time before that of the main verb. So their tense is relative, one could say. In this respect, there is no difference between the ways the participle can be used (e.g. "participle phrase" or "participle adjective", or whatever distinctions the various grammar books choose to make).

I have given you some general answers; to everything I said there are most probably exceptions and nuances.