I know from past experience that a correctly performed migration should never incur any downtime at all (providing you set everything up at the new provider, only then update domain's nameservers, and only in a few days cancel the old provider's account).
Apparently, their (email) account's setup is tied to MX records of domain's nameservers, making it impossible for a seamless transition.
For more information on correcting your MX records, see Why does my email account show that it is Pending Setup: MX Error?
And during this "afterwards" phase, the emails are rejected (as the domain is already pointing to GD, which is "pending" for some reason).
(Luckily I have an "extra" domain to play with before moving the big ones.)I'm not sure I understand the situation in detail, but you can have more than one MX record, with different priorities (likely, you can set this up at your current host via the control panel).
Exchange Server 2013 can be installed on either Windows Server 2008 R2 (SP1 or later) or Windows Server 2012.
Depending on the server roles you are installing the pre-requisites vary.