That's correct, in theory, a representative can attend on your behalf, but they would of course require your passport.
Same as, a representative can submit your 90 day report in person, or even submit an extension application in the event the applicant is handicapped or has disabilities.
That said, if you're travelling and staying in different Provinces, then you should be carrying your passport.
I don't think there is one answer fits all scenarios in this case as there as so many variables.
Provided you return by the under consideration period date given, then you can't be on overstay, but if you couldn't return until a later date then technically you would be on overstay.
I always apply 30-45 days before my extension expires.
My IO always date the under consideration period 30 days from the expiry date of my current extension.
They always call when the new extension application is approved, which is usually somewhere between the date my previous extension expired and the 30 day under consideration period, although it was once approved before my current extension had expired.
If you were planning to travel locally during the under consideration period then my advice would be to notify your IO of those dates, and provided you return by the end of the 30 day under consideration period it shouldn't be an issue.