Can you explain whether I do or don't need the possessive form, please?
I don't remember this photo's being taken.
The most recent point at which this migration could have taken place is 12,000 years ago, with the earliest period's remaining a matter of some unresolved contention
Top answer
The possessive form is still required by strict grammarians, but many now accept the popular usage of the objective form.
— Mister Micawber
The possessive form is still required by strict grammarians, but many now accept the popular usage of the objective form.
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The point is (or was) whether the speaker is referrring to the object or the attribute:
I like your dancing = I like the dance that you do. I like you dancing = I like you when you are dancing.
By the way, think the possessive sounds hypercorrect in your original examples; for one thing, the 's possessive is not generally used of inanimate objects