Building an AM receiver for 10Mhz (NIST WWV)
A lot of questions but if you are looking for a WWV receiver design here is one and below is one of the circuits contained on the link: -




Do I need any special length for my antenna?


YES but it's probably not too critical if you are receiving a decent signal. A quarter wave dipole at 10MHz is optimum but this will be 7.5m long so try a couple of metres.


Can any inductor/capacitor combo be used that meets the LC resonance
frequency equation? (C=100nF, L=50.3uH, should resonate at 10Mhz)


NO, you need to tailor the inductance capacitance ratio to give a decent Q factor but not be susceptible to parasitic components. Try for a resonant circuit that uses no less than 50pF - 50pF and 5060nH resonate at 10MHz - if you half C, double L for same resonance at an improvement in Q of 2:1. BTW it's "M" not "m" in MHz


I'm guessing this is tied to the output voltage or current of the
antenna. How would I calculate this?


Tricky, suck it and see


Would amplification between any step be useful for final input on my
Arduino? (Antenna->LC->RC->output->Arduino)


See design linked to for best guess at what you should be aiming for.


Would I run the output into another LC->RC circuit tuned to get the
sub-carrier? Does an LC->RC circuit even work for dual-side-band? (I
honestly couldn't find a difference between normal AM signals and AM
DSB)


WWV uses up to 100% modulated AM (or normal DSB) transmission. Use a diode detector (as per link) to produce the demodulated waveforms.