OPA2209: Noise Density

Brian,

1. The resistor on the input of your circuit itself will generate thermal noise that is directly related to the resistance value.  The precision labs video series has a section on noise that covers this and the other topics in this question in detail.  Amp Precision Labs.  Just look for the noise subsection.   
2. Current noise of 500fA/rtHz is typical for bipolar devices.  Current noise will translate into voltage noise when it flows through a resistor.  Keep in mind that the resistor itself generates noise.  Depending on your source impedance the current noise may not be a significant factor.  The point where the thermal noise is equal to the voltage noise from the current noise multiplied by the source impedance is about 65.8kohm for 500fA/rtHz of current noise.. See the details are covered in this application note: Impact of current noise.  CMOS and JFET devices have much lower current noise.  If you are using large input impedances you may want to choose a CMOS or JFET device (e.g. OPA827 JFET).
3. Bias current acts the same as current noise.  It translates into a voltage when it flows through a resistor.  The difference is that bias current is a DC signal and current noise is an AC signal.
4. When considering an amplifier to drive AD7606BSTZ, you need to consider the ADC noise.  In this particular case the ADC was designed with 1Mohm input resistors.  The large input resistors are required to attenuate the input signal and provide a high input impedance.  However, these large resistors are relatively noisy.  This noise level will be much higher than your amplifier noise.  The overall noise on the ADC is kept reasonable by limiting the bandwidth with an internal 23kHz filer.  This filter will also act on the amplifiers noise.  Long story short:  for this ADC you shouldn't worry about amplifier noise when the amplifier is in a buffer configuration.  The ADC noise is much higher than most amplifiers.  
5. Note: TI offers an equitant ADC: ADS8588.  This ADC has a similar noise performance so you shouldn't worry about amplifier noise for this option either.  For other ADC options that do not have the integrated 1M resistors you may need to consider op amp noise.
6. For bias current and offset, the calculation is simple:
   1. Ib x Rin = Vos_ib
   2. Vos from the amplifier is in the specification.
   3. Ib x Rin will add directly to Vos.  Vos_total = Ib x Rin + Vos
   4. The offset is a direct error on the ADC input.  The ADC will have an offset as well.  You should try to have the amplifier Vos_total equal to or less than the ADC offset.
   5. The offset on AD7606BSTZ is called Bipolar Zero Code Error.  It is 6LSB on +/-10V range, 16bits.  This corresponds to an offset of 1.8mV.
   6. The AD7606BSTZ offset is also large compared to what most amplifiers will have.

There isn't really a simple selection guide to help you determine when to use a particular amplifier.  You have to look at the specification of the ADC and the amplifier and calculate the errors.  Factors like source impedance will impact your result Amp Precision Labs and adc precision labs can help show you how to do the analysis.

I hope this helps.

Art