I use Beyond Compare when I need to copy files in bulk. There are many other similar products. I will use explorer if I am dealing with a few files via drag and drop... but if lots of files I prefer to see progress of the copy task via a good GUI.
It is possible to configure Beyond Compare to preserve timestamps...
Others use RoboCopy, which is a free utility... However, I am not familiar with its settings. You will find lots of tutorials on this.
GoodSynch also preserves timestamps...
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Duplicate Files....
I would be nervous to provide advise on handling duplicate files ... mainly because files with the same names in different folders may contain different content or may be previous versions of files, etc...
Copying From OneDrive.
I now have a personal policy of keeping zero files inside OneDrive and have removed it from my system... I have had too many issues ...
However, if I was copying files from OneDrive... I would copy the files to an external drive (USB fob or disk) ... because some of the one drive files might be tokens rather than real files.. it is prudent to leave the copy running for some time and then double check that all the files have travelled from OneDrive Cloud to your external drive. Copying to an external drive helps ensure that you are copying to a destination outside of OneDrive influence.
Only when I have all my files on an external drive would I consider deleting them from the OneDrive ecosystem.
You can log into OneDrive via a Web interface and review what files are actually in the OneDrive cloud.