thermodynamics - Calculation of internal energy and entropy of ideal gas

I am having some trouble with calculation of entropy and internal energy for ideal gas. All sources which provide such derivations which I managed to find either assume a priori that $U$ is some constant times $T$ or that $C_V$ is constant. I am now wondering if this calculation is even possible without a priori assuming one of these things, or in other words: is equation of state sufficient physical input to derive all thermodynamical quantities? So far I managed to show the following (basing only on EOS and principles of thermodynamics):

1. 
   

   $U$ depends only on T, $U=U(T)$. Similarly, $C_V$ only depends on $T$. Therefore $\mathrm d U=C_V(T) \mathrm d T$.

   
   


2. 
   

   U=TS-pV.

   
   


3. 
   
   

   $\mathrm d S=\frac{nR}{V} \mathrm d V+\frac{C_V}{T} \mathrm d T$.

   
   


4. 
   

   Integration of point 3. and using 2. I get that

   
   

$U=nRT( log(\frac{V}{V_0})+\int_{T_0}^T\frac{C_V(T')}{nRT'} \mathrm d T' -1)$,

where $T_0$ and $V_0$ are values in some reference state. But this formula seems to contradict 1.! Clearly, this expression cannot be expressed by temperature alone. What is wrong in my reasoning? Can I calculate $U$ and $S$ at all if I don't assume something more than EOS?

For reference of people who might have the same problem in the future I add proofs of points (1-3):

Ad 1. $\mathrm d S = \frac{1}{T}(\mathrm d T +p \mathrm d V)=\frac{1}{T}(p + (\frac{\partial U}{\partial V})_T) \mathrm d V + \frac{1}{T}(\frac{\partial U}{\partial T})_V \mathrm d T$.

This is an exact differential so mixed partial derivatives are equal. Using this and equation of state one gets the hypothesis.

Ad 2. This holds from first law because $U$ is extensive quanitity (formally one uses Euler's theorem about homogeneous functions). For a positive parameter $t$ we have that $U(tS,tV)=tU(S,V)$. Differentiating w.r.t. $t$ and then setting it to $1$ yields the hypothesis.

Ad 3. One just uses the first law, definition of $C_V$ and the fact that derivative of $U$ w.r.t. $V$ vanishes.