A couple of things come to mind.

When an email is sent by an smtp server the sender address will be that of the server. so smtp.somecomany.com will likely result in the sender being joebloggs@somecompany.com (event if they are not connected to somecompany.coms network).

Some overseas mail delivery (pop/imap etc) services will block incoming email if it is identified as a risk - so somecomany.com might be on the unrecognised list or marked as an Open Relay.

Authentication to an SMTP service as far as I know is not affected by who Internet service provider is. The credentials required are that required to connect to the SMTP server (which might be the same as those used to connect to the internet). Most these days will use TLS security with a different port (often 465) as pointed out in earlier posts.

If you are using Outlook you can have multiple accounts set up to receive email into the same mail folders. When sending email you get a little 'Send From' list above the TO: line that allows one to select. You can have one configured when overseas and one configured for when local. It is also easy to set the default.

As an aside.
I see a few recommendations with regards to using a cloud service. While these can be very good products to use, it is sometimes useful to consider the NZ Privacy act and how it might apply to a specific business before making such a recommendation.