What is the equation of state for a relativistic fluid/gas?
Your equations seem correct. Note, you can also get $v$ without having to differentiate, from $E=\gamma mc^2$ and $p=\gamma mv$. 


Here are some notes on relativistic fluids as related to stellar interiors, including derivation of these equations and their solutions in the non- and ultra-relavistic limits http://www.jb.man.ac.uk/~smao/starHtml/equationState.pdf


Correct, the integrals cannot be solved analytically (only in the two limits) so there is no general analytic formula