'It's' is not a possessive adjective or pronoun. It is a contraction of 'it' and 'is'. So, please, "the dog and it's puppy" would mean, if anything, that the dog was a puppy -or something like that. It would not mean 'the dog and the puppy of the dog'.
'Your' and 'you're', likewise, are not interchangeable!
Some of the other obvious ones (with the medal of honour going to 'I would of done') in Bob's post..