Can the radiator be replaced without changing the valves?

Sometimes, but it is poor economy. If the valve set is old, that is often what starts leaking after the new unit is fitted.

Does the new radiator output need to be recalculated?

Yes. You should not automatically fit something "roughly the same". Room size, system regime and the real output target all matter.

Often yes, but it should be agreed in advance. Heavy cast iron units or difficult removal can be priced separately.

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Heating radiator replacement in Israel — remove old, fit new, test the circuit | KABLAY