A update for you re the hotmail found to be causing the phone to getting hot and the battery draining fast.
With the help of Apple we found it was specifically down to the contacts part of the hotmail settings only with my account so must be a setting with hotmail on their end as my partners account had no issue on my phone.
When you switch off the contacts tab in the hotmail settings you loose your contacts but all is fine with the phone switch it back on you get contacts back but the phone soon heats up and battery drops like a stone .
I had only set up my contacts with hotmail in the first place as I understood that's what was required, so anyway i transferred all my contacts to icloud contacts and switched of the contacts tab in the hotmail settings and all is now perfect.
So my contacts are linked and are backed up in itunes and icloud and my hotmail e-mail/ calendar/ notes all work with hotmail. And the phone is cool and the battery lasts day and half easy.
I never persuade this with microsoft as it's easier to contact the dead than get through to them.
Again I hope this helps you out.