Mailboxes are already moved from on-prem to Exchange online by Hybrid MX still points to on-prem. Customer want to keep on-prem Exchange server to do some administrative tasks later, but they'd like to move it to Azure Data center.
1. Can they move on-prem exchange to Azure data center and then point the MX record directly to Office 365? 2. If they can move on-prem exch to Azure, do they require on-prem EDB also to be moved to Azure data center to avoid any functionality issues (or)
Is that okay to get rid of on-prem EDB and move just the on-prem server? If we get rid of on-prem EDB will it affect certain functionalities for moved users?
Once the mailboxes are moved to Office 365, On-Premise Exchange will convert them to Remote Mailbox. If the mailboxes are migrated to Office 365, and local databases are empty then no point keeping database files (EDB) as you're not using them. But properly delete the databases after confirming they no longer holding the any mailbox then clean the database and log files.
It is recommended by Microsoft to keep one Exchange alive for admin tasks but not sure if you can take local server out there in Azure and co-exist with EXO. That doesn't seems possible.
Go through some blogs by Official MS or may be you can contact Apps4Rent` Azure advisory team for consultation.