I realise this is probably a somewhat confusing question, which really has two distinct parts, but I thought of them in relation to one another so...
Firstly, with the phrase 'Having done that was a rewarding experience', is the word having acting as gerund or a participle (i.e. (My) having done that was a rewarding experience)?
Additionally, I was curious as to how to categorize the italicised words in phrases such as the ones below.
He paused, weighing up his options, then chose. The man, considering his options, looked into the distance. Having thought it over, I think it's a good idea.
What grammatical term describes these words? I would understand, for instance, if a phrase said 'He paused, weighed up his options, then chose', since 'weighed' is clearly a past participle describing how 'he weighed up his options'; however, 'he weighing up his options' is clearly nonsensical. Are they simply verb phrases with no relation to the subject? Perhaps there is an obvious answer; hopefully someone can provide it or at least offer some improved grammatical context to my question.