You can find a good program to estimate loop inductance and some other
parameters on Reg Edwards' web pages. He has a link in many of his
postings on this group and the r.r.a.antenna one. The impedance comes
from the Q and the fact that you are resonating it--or at least it's
presumed that you are resonating it. So if it has a Q of 300 and the
inductive reactance is 50 ohms, the resistance when resonated is
300*50=15000 ohms, for example. That's why folk like to use preamps at
the antenna: transform that high impedance down to a low impedance
that's easy to send along a transmission line. Seems to me that if
they are having trouble with intermod distortion in the preamp, the
preamp isn't designed properly. It's not terribly difficult to get
very low distortion at LF these days. By the way, if you build a
really big loop and have so much signal you can attenuate it, that
gives you a chance to lower the Q and increase the bandwidth: if what
you want to listen to occupies much bandwidth, you don't want your
antenna to filter out the information you want to listen to!

I'd suggest you read an antenna book like Johnson and Jasik, or the
antennas chapter of King, Mimno and Wing's "Transmission Lines,
Antennas and Waveguides." They will make it a lot clearer why you
might want a balanced loop. You don't need a grounded center-tap to
make it balanced--just make it very symmetrical.

Cheers,
Tom