Source: https://github.com/osTicket/osTicket
Very customizable if you know how to code. Old and outdated UI but can pay for 3rd party custom themes or plugins that give it a fresh coat of paint. They are currently in a complete code rewrite for v2 and that has been in "progress" for over a year with no ETA from the devs. People on reddit think the project might be dead or dying.
There is no time tracking. There is a client portal built-in. You can add scheduled task with this: https://github.com/Kyrillian/osTicket-TaskScheduler