Entropy of the Sun 
*

*Is it possible to measure or calculate the total entropy of the Sun?

*Assuming it changes over time, what are its current first and second derivatives w.r.t. time? 

*What is our prediction on its asymptotic behavior (barring possible collisions with other bodies)?

 A: For an ideal gas, you can always relate the entropy to the pressure and density via the relation*
$$
S(r)\simeq P(r)\rho^{-\gamma}(r)
$$
where $\gamma=5/3$ is the adiabatic index. Reasonable (and simple) estimates exist for the density and pressure profiles (e.g., $\rho(r)\simeq\rho_c\left(1-r/R\right)$ where $R$ is radius of the star and $\rho_c$ is the central density).
I do not know that there are functional forms of $P(r,t)$ and $\rho(r,t)$, so I do not know how to answer the questions about the time-derivatives of $S$. Perhaps there is a stellar structure code online that you can modify to give that for you?

*This definition stems from the Stackur-Tetrode relation of the entropy density,
$$
s=C_v\ln\left(P\rho^{-\gamma}\right)+{\rm const}
$$
but eliminates the "clunky" $C_v$ and ${\rm const}$ terms.
A: The entropy of the sun is roughly $10^{35} \mathrm{J} /\mathrm{K}$ (see http://dx.doi.org/10.1103/PhysRevD.7.2333 where the entropy of the sun is given as $10^{42}\mathrm{erg}°\mathrm{K}^{-1}$) 
It can be calculated using Boltzmanns law ( $S = k_B \ln{W} )$ where $W$ is the disorder parameter of the sun (depends on the number of atoms in the sun). 
I did not get you second question.
