Can my year start in April or September, like a fiscal or school year?

Yes. Open the calendar menu and choose Year Range, then select Fiscal under Mode and choose any start month. This suits fiscal years, school years, and project cycles, and a fiscal year can be labeled by the year it ends.

If you only want the next 12 months from today, use Rolling instead. Year Range is set per calendar, and only the calendar owner can change it.