Do you pay for a domain? They likely provide dynamic DNS (DNS). If you’re lucky, they have an API for it, instead of an app, and you can configure a cronjob on your home server to run every 1-5 minutes (or more often, if your IP is super unstable!).

I have very similar requirements, but I’m currently using a Pi with some external drives since that’s how I started out. Would you mind sharing what you ended up buying? My place is pretty small, so the ‘quiet’ requirement is one I care about a lot. Personally I’d love to get something passively cooled, but I haven’t seen much!