I host Synapse using this playbook. I can highly recommend it - the instructions are very clear and detailed and ongoing maintenance is straightforward too (just git pull and redeploy, and 5% of the time modify a deprecated variable).

As for how to delegate to a subdomain - that’s covered here. Basically - you set it up on matrix.example.eu and then have a “well known” file hosted at example.eu that tells other clients/servers where to look.