grammar - What do you call the grammatical element that describes the state of being of the subject?

The first four sentences start with what the Oxford Dictionary of English Grammar (p291) calls a participle clause:

A non-finite clause with an -ing form or -ed form as its principal
verbal component. (Also called participial clause.) Examples:

• Looking to neither right nor left, he marched out




• Having been warned before, he did not do it again.

Sentence 5 does not have a participle clause, and is in fact ambiguous. I suspect that most readers would interpret happily, not as referring to the state of mind of the he taking the advice, but rather as expressing the writer's attitude to the fact that the he took the advice. This type of adverbial is called a disjunct (ODEG, p124).

To disambiguate the sentence happily needs to be repositioned: He happily took my advice or He took my advice happily.

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As an aside, a participle clause that has a different (implicit) subject than the subject of the matrix clause is called a misrelated, dangling or hanging particple. One of the ODEG's examples is: Sleeping in mine orchard, a serpent stung me (Hamlet, 1.5).