Can I get reimbursed for a broken controller upon opening a new Xbox?

Since it's brand new, start with the retailer you purchased it from to see if they can exchange it. That's usually quicker. Otherwise, it would be under warranty so you can request a replacement. https://support.xbox.com/help/hardware-network/controller/controller-warranty

Looks like your joystiq might be broken or out of its place you can try to open it and fix it, go to a place that fix controllers or buy a new one...hope that helps!!

The jack on the controller is without a doubt broken what i can refer to you is going to gamestop and getting a warranty on a controller it only costs 10$ extra and the warranty lasts a year so anytime u have this problem u can bring it back and u will
only pay 10$ for a new warranty when u get the new controller because its a new controller so new warranty

My X1 headset adapter stopped working on 10/27. No mute light and no mic getting to Xbox. The mic works, I can hear it in the headphones, but it's not getting through adapter. I tried updating controllers, but they didn't update.

I went out and bought another X1 headset adapter, controller auto updated, new adapter works fine, old one still doesn't. Too soon for this stuff to brake down. I want warranty or reimbursement to kick in here. Help!

The new Elite 2 controller seems to have answered the past issues. To get your controller serviced:

https://support.xbox.com/en-US/xbox-one/accessories/wireless-controller-broken-wont-work