liquid polymers of high molecular weight We often see a limit of melting point as a function of polymer chain length. Polyethlyene tops at about 190C, and polyethlyene glycol saturates at only 67C no matter how long the chain is:
http://www.sigmaaldrich.com/materials-science/material-science-products.html?TablePage=20204110
Although it takes more energy per molecule to slide with respect to it's neighbors and be able to move, it appears that the number of degrees of freedom (and available thermal energy per molecule) increases proportionately.
How does viscosity scale with molecular weight? My reasoning: Viscosity ~ size because larger molecules would have more relative velocity with respect to each-other. Size would be ~ length^(1/3) if they tended to coil randomly, or ~length if extensive shearing leads to chain straightening. Also, polymers tend to be non-newtonian; would the non-newtonian-ness of the fluid show a similar scaling behavior?
 A: Not sure about the pure polymer viscosity, but here are some data on the polymer solutions.
The viscosity of even dilute polymer solutions is usually far larger than just the viscosity of the background solvent, due to the large differences in size between the polymer and solvent molecules. The contribution of polymer to overall viscosity is measured by dimensionless number: intrinsic viscosity.
From wikipedia's Mark–Houwink equation

The 'Mark–Houwink equation', also known as the 'Mark-Houwink-Sakurada equation' or the 'Kuhn-Mark-Houwink-Sakurada equation' gives a relation between intrinsic viscosity $[\eta]$ and molecular weight $M$
  $$ [\eta]=KM^a$$
  From this equation the molecular weight of a polymer can be determined from data on the intrinsic viscosity and vice versa.
The values of the Mark–Houwink parameters, $a$ and $K$, depend on the particular polymer-solvent system. For solvents, a value of $a=0.5$ is indicative of a theta solvent. A value of $a=0.8$ is typical for good solvents. For most flexible polymers, $0.5\leq a\leq 0.8$. For semi-flexible polymers, $a\ge 0.8$. For polymers with an absolute rigid rod, such as Tobacco mosaic virus, $a=2.0$.

The theoretical analysis which reproduces this equation was done by Flory and Fox. Here we read

One of the most successful models comes from Flory and Fox who modeled the random coil as a series of “beads on a string” or a “jointed chain”.
  The string is flexible, but beads are rigid. Flory and Fox considered that hydrodynamic friction causes the solvent near the center of the molecule to move with the same velocity as the center of mass, but solvent near the edges is free to flow into and out of the molecule. This led them to a relationship between the intrinsic viscosity and the mean square radius of the polymer chain in a theta solvent. Their model is:
  $$
[\eta] = \Phi_0 \langle r^2 \rangle^{3/2} / M
$$
  where $\langle r^2 \rangle$ is the mean squared end-to-end distance of the chain, and
  $\Phi$ is a universal constant having the value $2.87\times 10^{23}$. In practice, this
  constant varies somewhat from polymer to polymer with an experimental
  value closer to $2.5\times 10^{23}$.

The original work by Flory and Fox:

Flory, P. J., and T. G. Fox. "Treatment of intrinsic viscosities." Journal of the American Chemical Society 73.5 (1951): 1904-1908. DOI:10.1021/ja01149a002.

