Dave,

Assuming the 100uH is a standard ferrite rod type antenna, you can
likely give it a boost by winding just a couple turns (experiment: try
from one to five turns, for example) of wire around it, grounding one
end of that wire and extending the other end into a wire antenna
several feet long, up in the air as best you can. I'd suggest ten feet
minimum length, on up to maybe a hundred feet, but just do what you
can. You can connect the circuit common to a "ground" made up of a
ground rod driven into the earth, or just a wire on or close to the
earth. Experiment! You're not likely to hurt anything, so long as you
keep it away from power lines.

Dunno for sure about the background whine. Is it independent of
tuning, or is it only there when listening to the station? Does the
pitch change with changes in the tuning? Can you describe it better?
It could be an oscillation in your circuit. Substituting the 12k for
10k should not be a problem. But I'd add a bypass capacitor across the
battery voltage (and keep all the leads reasonably short) I'd use
maybe 10uF at 16V or more, be sure the polarity matches the battery,
and put it physically near Q3.

In troubleshooting things like the whine when you have limited test
equipment, you can just try things like opening connections in the
circuit to see which part of the circuit it's in. For example,
disconnecting the 100n capacitor should tell you if it's a problem in
the input section or output.

Good luck, and don't be afraid to experiment.

Cheers,
Tom