whats standard is if you have a grace period and the phone is defective, in this case this one seems to be based on your comments, standard practice for most is to simply take the phone back and exchange it or get a refund. you should not have any of these issues you are describing on a new out of the box untouched device.
the shop sounds seedy to me based on what you wrote, and i can only go on what you wrote so its just speculation on my part. clearly you have a relationship with the shop so thats all that matters.
still though, the customer is always right and if they want you to jump through hoops and try and prove something, that's not good customer service. I mean you have better things to do than to make false claims about issues with a "new" device, no?
bottom line, not much anyone can do to help you over the internet. if a factory reset doesn't work, i strongly suggest you take it back for a refund. dont get suckered into a faulty device.