How to use my bits to transmit this photodiode signal in the most efficient way possible?
To get started, the NEP is the power of the signal that give you an SNR of 1 (not very useful for communication) at a bandwidth of 1 Hz (Damn slow). To do a quick back of the envelope calculation, pick a bandwidth, close to half your desired data rate, and a desired SNR, and get \$P = SNR \cdot NEP \cdot \sqrt{BW}\$. You will probavly get something higher than 1 pA.

First of all, does it make sense that my signal is smaller than the Dark Current?
The current you will measure will be the photo-current (your signal) plus the dark current. Be careful, just like the photo-current, the dark current is susceptible to shot noise, and this is ignored by the NEP calculation, so if you are detecting signals of the same order of magnitude as your dark current you should look more closely at that. Also, you will need a trans-impedance amplifier to convert from current to voltage for your ADC. This will come with noise and bandwidth considerations too.

Now, assuming this is the case, what should I do with my ADC in order to have all my bits carrying information from the signal?. Most communication systems use balanced signaling, for example 8bits-10bits encoding, and then use AC-coupling to remove the average of the signal. The signal from your ADC is now symmetrical, around 0V or what reference you used to bias the ADC input. Background light is less of an issue. If you want to stay DC-coupled because of your modulation, for example with Pulse Position Modulation, it's very likely that you will need to estimate the low and high level of your signal with digital signal processing to get a reliable system.