The water heater burst and ruined my things. Who pays?
Your policy pays for your belongings; the owner replaces the water heater and repairs the building. Those are two separate matters that people expect to be one. If the owner was negligent, your insurer may pursue them afterwards, and that happens between insurers rather than between you.
Is it covered if it had been leaking for a while?
Gradual damage over time is generally excluded, so a sudden burst and a long-running seep are treated very differently. In an older rental you are the person most likely to spot the early signs, and reporting them to the owner protects you as well as the building.
Do I have to move out while it dries?
Sometimes, and if a covered loss makes the unit uninhabitable, loss of use pays the extra cost of staying elsewhere. On this kind of claim that is frequently the largest part of what the policy pays.