Is a part of the spinal cord from the mesoderm?
The caudalmost part of the neural tube is formed by the caudal eminence. Isn't this mesodermal? 


No,the neural tube is not mesodermal at caudal end.

According to Gray's Anatomy:


  At the time of caudal neuropore closure, the caudal midline cells are generically termed the caudal eminence (O’Rahilly and Müller 2004). A specific population of mesenchymal cells, the caudoneural hinge or junction, shares the same molecular markers as the primitive node. These cells aggregate at the midline and undergo mesenchymal/ epithelial transformation, producing a cellular cylinder that is contiguous with the caudal end of the neural tube. Further elongation of the caudal neural tube involves cavitation of the neural cylinder.


So, the cells in the caudal eminence form a solid cylinder(medullary cord) which is made hollow by elongation of neural tube. Therefore, mesodermal cells of caudal eminence are not forming the caudal end of neural tube.

Let's see the process of neural tube elongation from the citation given in the Gray's textbook:


Abbreviation used can be seen here

Schematic diagram can be seen here