Not really an issue - just use something like NoIP. No need to pay Amazon or Google for anything.
To send email you need a static IP with correct reverse DNS, or other people's servers will reject your mail (best case) or silently mark it as spam. Welcome to the real world of email deliverability, the worst part of running your own mail server.
On the receiving end I use a super inexpensive spam filtering service too, MX Guard dog. If my IP suddenly changes then it queues up mail until host resolution succeeds again.