Can the Contractor proceed with the Contractor’s Design works before formal acceptance by the PM as the works will affect other subsequent works? What should the PM do?
Contractor can proceed without design acceptance but it would be at their own risk. If it was subsequently rejected for a valid reason under the contract then any cost incurred would be down to the Contractor along with any time implications.
If the Contractor proceeding with the works could affect anything else then the PM can always instruct the Contractor to stop the works which they would have to obey, which would NOT give rise to a compensation event either.
How about if the Contractor does not proceed with the works (before acceptance) immediately, then it will affect the subsequent works. What should the PM do?
Not quite sure what you mean by your question. If they do not proceed with the works as the design has not been accepted but remaining works could be affected - you could call a risk reduction meeting to discuss the potential impact and agree a way forward which may or may not require an instruction. Any liability that results will be assessed on its own merits in accordance with the contract.