Can you use "the" with proper names in the posessive case?
For example: can I say this is the Lady Mary's car or this is ___ Lady Mary's car?
This use of the article started falling into disuse centuries ago, and there's been a huge shift towards the article-less form in the last couple of decades. You should only include the article today if you're trying to evoke "medieval" associations.
Note that the Lady XXX was only ever "valid" to refer to a titled woman (awarded the status Lady XXX by some recognised civic ceremony). You certainly wouldn't use it of a "commoner".