Migrating to New Hosting

Hi there,

I'm using Office 365 and I have a custom domain in my account. (winecellarspec.com)

Planning to transfer  to a new hosting, this includes changing to new DNS setting.

Domain is on Godaddy. Hosting provider on Hostgator.

Basically after the new DNS has propagated. I'll be adding those DNS records generated inside Office 365 both CNAME and MX.

Now my question:

Is that possible without a problem? Do I need to verify my domain again?

Will I get downtime with my emails? 

Will the users be remove? How about Calendars? Contacts? Will those be affected?

Please advice the best course of action how to migrate. 

Thanks!