Finding the time domain equation corresponding to a given DFT
(1) I'm not sure my intuition is right 


Almost, but you're missing the fact that if you're sampling at 4000kHz, the signals alias; after sampling at 4000kHz, a sine wave at 1500Hz is indistinguishable from one at 2500Hz or 5500Hz, etc.  Also, because the work is being done in complex numbers, negative frequencies have meaning: $e^{-j \omega t}$ is different from $e^{j \omega t}$


  (2) I don't know how to prove it


Go back and study the Fourier Transform more.  You want the Euler identity tattooed onto your frontal lobes: $e^{j \theta} = \cos \theta + j \sin \theta$.  You also want the two offshoots of that: $$\cos \theta = \frac{e^{j \theta} + e^{-j \theta}}{2}$$ and $$\sin \theta = \frac{e^{j \theta} - e^{-j \theta}}{j2}$$

prove these to yourself.  It'll help you out immensely later.

So (after you review the Fourier transform) you have a signal with a real part and an imaginary part: $$\sqrt{32} \left ( e^{j 2 \pi \frac{3}{8}n} + e^{j 2 \pi \frac{5}{8}n} \right) - j\sqrt{32} \left ( e^{j 2 \pi \frac{3}{8}n} - e^{j 2 \pi \frac{5}{8}n} \right)$$  Note that $2 \pi \frac{5}{8} n = 2 \pi (1 - \frac{3}{8}) n$ and recall that $e^{j 2\pi + \theta} = e^{j \theta}$; after some clever trig identities do these look like anything I've just shown you?


  (3) I don't see how to calculate the amplitude A0


It should drop out of the calculations above.


  (4) and finally, is finding x(t) really the answer to the question of writing "the time-domain equation for the discrete () sinusoid in trigonometric form"


Yes, but mind your aliasing!