(1) What is the criterion of calling one signal as real and the other imaginary from the two orthogonal coils
Assuming that you have detection axes aligned with the $x$- and $y$- axes ($y$ being offset by $+90^\circ$ from $x$), then for a single spin with precession frequency $\omega$, the signals you get will be:
$$S_x = \cos(\omega t); \quad S_y = \sin(\omega t),$$
and then we choose to define the complex signal as
$$S := S_x + \mathrm{i}S_y = \exp(\mathrm{i}\omega t).$$
In this case $x$ is the real axis and $y$ is the imaginary axis. Complex FT of this yields a peak at frequency $\omega$, as desired. However, this isn't the only way to do it. If we wanted to swap them around, we chould choose $y$ as the real axis and $x$ as the imaginary axis. However, we would need to be careful to include a minus sign in the definition:
$$S' := S_y - \mathrm{i}S_x = -\mathrm{i}\exp(\mathrm{i}\omega t)$$
This is the same signal as before but with an additional global phase factor, which is unimportant. Notice that if we defined $S' := S_y + \mathrm{i}S_x$ we would get a peak with negative frequency. In some sense, the imaginary isn't the $x$-axis, it's the negative $x$-axis. I think the summary from this is that the imaginary axis should be offset from the real axis by $+90^\circ$ (and not $-90^\circ$).
(2) Which FID is used for displaying the NMR spectrum?
Both, actually: we don't FT only the real or imaginary part of the FID. We form a complex FID and perform a complex FT on that, i.e. $\int_{-\infty}^\infty f(t)\exp(-\mathrm{i}\omega t)\,\mathrm{d}t$ to get a complex spectrum.
Traditionally, only the real part of the spectrum is displayed to the user, though.
[...] Both FID from the two receivers should appear like a free induction curve since are both real experimental values. Discrete Fourier transform of each should yield the same frequency information i.e., the same NMR spectrum. There must a be phase difference but frequency information must be the same. Is this correct?
Yes, both real and imaginary components of the FID will contain the same frequency information. However, neither component alone is sufficient for providing quadrature detection, because one signal is
$$S_x = \cos(\omega t) = \frac{\exp(\mathrm{i}\omega t) + \exp(-\mathrm{i}\omega t)}{2}$$
and the other signal is
$$S_y = \sin(\omega t) = \frac{\exp(\mathrm{i}\omega t) - \exp(-\mathrm{i}\omega t)}{2\mathrm{i}}$$
so if each of them is individually FT'd, you get duplicate peaks at $\pm\omega$, hence the need to combine them. (If your frequencies are only positive, then of course you don't need two components; you just discard the negative frequency parts, or equivalently use a cosine transform instead, i.e. $\int_{-\infty}^\infty f(t)\cos(\omega t) \,\mathrm{d}t$. But in NMR, the frequencies are always going to have both signs.)
As for the spectra, you can reconstruct the imaginary part of the spectrum from the real part of the spectrum via a Hilbert transform. The resulting formulae relating the two parts are also known as the Kramers–Kronig relations.
