Can the radiator be reinstalled on the old seals?

Better not. The saving is small, while the risk of leakage after reassembly is higher.

If the wall has been replastered, do the brackets need to be reset?

Often yes. New plaster, tile or panels change both level and depth.

Yes, but the pipe ends must be sealed properly, and the radiator itself should be stored dry and safely, especially if it is heavy.

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Radiator removal and refit in Israel — temporary dismount and reinstallation | KABLAY