Yes, big hotels organize Christmas Lunches or dinners, and believe me, there are many many hotels overhere! :)
Yes, of course you can bring your dog here, it would be difficult to explain everything now, so just see the Pet and Animals Group here in Qatar Living, and you will find the answer for all your queastions.
I am sure you will be able to find jobs. You can work with big companies in the HR department, but about teaching english I don't know much, but sure there is a need for it, as it's the second most important language here (or maybe the first).
Welcome in Doha in advance!