This is more anecdotal than a specific answer, but I was once invited to spend a few days at Cambridge as the guest of one of the colleges. I otherwise have no association with them. For dinner (hmmm and sherry hour), however, I was asked to rent an appropriate academic gown, which I did.
So, it seems that in those circumstances when others, including professors and students, would be gowned, it would probably be fine, even required, that you do also.
So, your first bullet seems to be a clear "yes". The second would depend on the traditions in the place. So, probably not in the US. The third is probably acceptable nearly everywhere, if seen to be a bit odd. But if the kids are graduating from your old place then likely yes, otherwise see point two.
In the US, doctoral students are encouraged to purchase a gown, though rental is usually available. They then wear those at other formal academic events such as future graduation ceremonies and convocations and, perhaps formal dinners. But, customs vary. In the UK it was once common to see academics gowned (for lecture, say). Perhaps it still is. But in the US it is rare and requires a special occasion.