Choose the dimensions of the resistantors in this op-amp circuit with negative feedback to have a gain of 10
You know there's a problem since your resistor value for R2 is lower than that from an ideal op-amp calculation, and we know that with non-infinite gain it must be higher than the ideal value of 50K. 

In your second from last line, the gain is -10, not +10. 

I get: 

\$R_2 = \frac{R_1\cdot G (1+1/A_0)}{1-G/A_0}\$ where G is the desired gain (times minus 1) and \$A_0\$ is the open-loop gain of the op-amp.  

The resistor R3 is to eliminate offset caused by input bias current. If the input bias currents are equal, then making R3 = R1||R2 will do that.