Forget about combining with other “elements.” It’s meaningless. Any verb, and any participle, can be attended by any number of other elements, such as subjects, objects, adverbs, prepositional phrases, etc. You need to adjust your mental framework. Useful categories are “main” vs. “subordinate.” Participles are always subordinate, and their case, like their number and gender and their tense, depends on their relation to the main verb (more strictly, the leading verb, but never mind that).
The force of a participle (as conditional, concessive, causal, temporal, or merely circumstantial) is not built in to the participle. It depends on the reader’s understanding of the context. But introductory particles (e.g. καίπερ), grammatically optional, sometimes help to narrow it down, as in your ex.9 sentence.
In addition to participles, there are subordinate clauses, introduced by conjunctions such as επει, εἰ, ὄτε etc.—like English “although,” “while,” etc. These have finite verbs, just like main clauses. But where English uses subordinate clauses, Greek often uses a participle instead. “Although he is reporting many things” may be expressed either by a subordinate clause (ει και πολλα αγγελλει) or by a participle with or without καίπερ. If the latter, the form of the participle (nom., gen., whatever) depends on its relation to the verb of the main sentence.
But as to your question: yes, there is one other absolute participle construction. Impersonal verbs (e.g. δεῖ, ἔξεστιν, ἔδοξεν) use an accusative absolute (e.g. δέον, εξόν, δόξαν, always neuter), not a genitive absolute. By their very nature they can have no grammatical relation to the main sentence, hence they’re “absolute." —But you’re getting ahead of yourself.
If any or all of this is confusing, just ignore it. And If seneca has anything to say here, you should listen to him.