I can't give you a definitive answer but the Goldolphin Estate was taken over by the National Trust in 2000. Major safety work was undertaken on the extensive remains associated with Great Work and Godolphin mines, including the building of Cornish hedges around more than 20 open mine shafts and consolidation of the pumping engine house and stack at Leeds' Shaft. West Godolphin Mine is a grade 11 listed building and part of the Historic Environment so I would offer a tentative yes.
Part two of your question is best answered by a quote from the archives of 2003.
"The completion of a three-year landscape project at the historic Godolphin Estate was celebrated last week. The National Trust, which bought the 550-acre estate in January 2000, has undertaken extensive landscape and access improvements in a bid to make the site more visitor-friendly, both for tourists and local people. Footpaths and stiles throughout the estate have been improved - more than six miles of new paths created and a new estate guide produced. An education base has been installed at the renovated Godolphin Count House and there have already been some 850 visits by school groups. The trust has produced a new education resource pack for teachers, which helps provide self-guided tours of the estate. Traditional hedges and boundaries have been repaired and modern aluminium gates and concrete posts replaced with timber and granite. Major safety work has been undertaken on the extensive remains associated with Great Work and Godolphin mines, including the building of Cornish hedges around more than 20 open mine shafts and consolidation of the pumping engine house and stack at Leeds' Shaft. Parking areas have been created for visitors - the car parks surrounded by earth banks and landscaped so that the vehicles are camouflaged. Alastair Cameron, from the National Trust, said: "The project's extensive programme of improvement to this unique and historic landscape has involved a tremendous amount of hard work and dedication. "It is good to have an opportunity to thank everyone involved and celebrate their achievements, as well as paying tribute to those who helped fund the project." The completion of the Godolphin project is an important milestone in a unique partnership which developed as a result of a creative collaboration between the National Trust and the Schofield family. Two substantial legacies and grants from a number of funding bodies enabled the trust to buy the estate from the Schofields, who still own the historic Godolphin House and garden - as they have done since 1937".
The past is a foreign country: they do things differently there.