Self-Hosted Ticketing & Email Integration

I need an open-source self-hosted ticketing system installed, configured, and ready for daily use integrated with emails (Microsoft Office 365) with following goals:
1. 1. Any new incoming email instantly opens a ticket.
2. Replies on the same thread must append to the original ticket, while different subjects spin up fresh tickets.
3. Tickets should route themselves to the correct internal agent or group based on predefined rules.
4. The solution must recognize and process mail sent to multiple addresses across the various domains we use.
5. Should be branded, should not give third party logos, or watermarks
Emails used are hosted with Microsoft 365 (Primary), third-party hosting company and gmail

Preferred stack
I have experimented with Zammad and Frescout; Zammad (latest stable release) looks promising, yet I am open to any other open-source system that can meet every requirement above.

Working style
Because the server is on-prem, credentials cannot be shared. All installation and configuration will happen during a live screen-sharing session that we will schedule together.

Deliverables
• Clean installation of the chosen help-desk on our server
• Full email integration with Office 365 and external domains
• Rule set or scripting that enables automatic ticket assignment
Branded with company name and logo, Should not give third party logos, or watermarks
• Brief hand-over guide so we can add new agents, addresses, or rules ourselves

Once the platform runs smoothly in daily use and the guide is handed over, the project is complete.