I currently have my small business's mail hosted with an external provider, which is fine, however they only offer 5 emails under the current plan. I don't want to pay extra for an additional 5, and I have a Synology NAS that can run a mail server quite happily. As the additional mail addresses are going to be used for non-critical staff, is there a way to have the extra mail addresses hosted by the internal mail server, and the existing mail addresses hosted by the external provider? Thanks.
If you've already exhausted the 5 addresses/mailboxes (you don't state which) they gave you, then you can't add more addresses/mailboxes with the current setup.
In case they do allow you to add more addresses (not mailboxes), you could forward new addresses to your own server, which needs to have an MX or an A record in order to receive mail.
If not, there is no easy solution; you'd have to register a domain, set up your own mail server, and forward the old addresses to the mail provider (or not; no idea why you'd want to do that if you have to run your own mail server anyway.)
I would strongly advise against running your own mail server unless there is no other solution available - and there obviously is: just request more addresses from the mail provider.
Yes, e-mails and mailboxes are different entities, you have to elaborate, 5 "what" they provide, and can you have (and manage) "virtual" e-mails without separate mailboxes
No, each location inside domain have one final MX destination.