Can anybody help me out with finding the loop transfer function, the resonant frequency and the resistor value questions for the following circuit given the values?

 

You need to start show as your work.
As a tip notice that this RC and CR network forms a bandpass filter.
And for ωo = 1/RC ≈ 1.6KHz the phase shift is 0° but the "gain" is equal to 1/3. So to satisfied osculation condition R1 need to be equal to R1 = R2/2 = 5KΩ

 

You need to start show as your work.
As a tip notice that this RC and CR network forms a bandpass filter.
And for ωo = 1/RC ≈ 1.6KHz the phase shift is 0° but the "gain" is equal to 1/3. So to satisfied osculation condition R1 need to be equal to R1 = R2/2 = 5KΩ

How would you go about finding the loop transfer function, could you show the steps?

 

How would you go about finding the loop transfer function, could you show the steps?

That's not how it works. You show us your steps and we confirm your result or guide you in the direction of figuring it out for yourself. That way you will remember the method when the test rolls around. A hint would be to read the wiki on Transfer Functions.

http://lmgtfy.com/?q=Transfer+function
http://lmgtfy.com/?q=Loop+Transfer+function

Read the articles, and follow the links to make sure you understand the meaning of "loop transfer function", "open loop transfer function" and "closed loop transfer function".