The history of the Tessar in for example the wikipedia page answers some of your questions:
https://en.wikipedia.org/wiki/Tessar Basically, although the Tessar diagram looks a lot like a Cooke triplet with the rear singlet turned into a doublet, the two designs actually were derived independently from a common ancestor.
The designs are fairly similar in that if you swapped the rear element of one for another, it should make an image, but the aberration corrections would all be wrong. Lens designs have to be recomputed for the specific glasses available to the manufacturer, so that there is no single "Tessar" type, but a family of small variations. Meaning that even swapping the rear element of a Yashica Tessar-type with a different Tessar, say a Rollei or Mamiya Tessar-type of the same focal length, you would probably get a lens that focuses on-axis but has spherical aberration, and off-axis has astigmatism, coma, etc.
I have a Yashicamat EM, which is close to the LM, with a Yashinon lens. I looked at mine and the rear element is a doublet - you can tell by shining a flashlight in and looking for the faint reflection from the glass-glass interface (watch how the different reflections move in diff directions due to the curvature of the surfaces. Edit: There might be some Tessar variations where the surfaces have different curvature sign than the usual, I'm not sure).