You need to use styles of managing work that reflect the nature of the work and the responsibilities of the people doing the work. "Sprints" work okay for prod

You need to use styles of managing work that reflect the nature of the work and the responsibilities of the people doing the work. "Sprints" work okay for product work. I use them a lot less rigidly than most people do with success. On-call activities should be recorded in a kanban-style project, these things are made for rapid reprioritization. Whoever works that board should be dedicated to only that board; making members of a team work in multiple project management paradigms at the same time usually has bad and confusing outcomes. Lastly, the "support" tickets can be varied. If they're requests that are gated by your team, I'd add them to the on-call kanban board. If they're just question/answer I wouldn't bother tracking them, you'll more or less end up recreating ITIL. Some additional context would be to dedicate people to these boards for periods of time. Have a rotation that incorporates everyone doing a set amount of on-call on a recurring schedule. Build up a handoff procedure for the kanban board since context will be fresh every week due to the nature of the work.