Solar Battery Charging
This answers many but not all queries. Look at this and ask re what is still needed.

These SE questions which I have answered are liable to be useful

As others have said a series parallel arrangement (1S + 2P)is not at all good.
Capacity is that of the single cell. 


  I have a 3-AA battery holder and I am looking to build a solar battery charging circuit. I am planning on using 3 AA NiMh 1.2V batteries to be placed in the battery holder.


At least consider using a single Lithium Ion cell (or LiFePO4). These will prove easier to manage well. You'll need a control IC made for the task but thgey are low cost and make the charging task very easy. There is NO REALLY good way to solar charge NimH with small solar systems (Ask me how I know :-) ). 


  I am planning to use a 1N5817 blocking diode, so current does not get drawn back to the solar panel.


1N5817 is good. Except where panel voltage is critical (see below) a std 1A silicon 1N400x is fine.


  If the batteries are placed in parallel, then the total voltage for the battery pack will be 1.2V, 2300mAh * 3 = 6900 mAh. I have concerns about losses at such low voltages ?


Vseries = V1 + V2
mAh series = mAh of lowest if non identical.

Vparallel = MUST be the same (some exceptions but don't do it)
mAh parallel = sum of two


  I could also wire two of the batteries in parallel and 1 in series for 2.4V, 4600mAh. But I would have to balance the cells somehow ?


Very bad idea.
As above, mAh = that of lowest series item = here 2300 mAh.
No advantage in doing this.


  According to this page: http://www.amazon.com/Sunnytech%C2%AE-100ma-Module-System-Charger/dp/B00HQ9CUMO, it says that for a storage battery of 1.2V, you need a solar panel that outputs 2-2.5 V, and for a storage battery of 2.4V, you need a solar panel outputting 3.5-4V.


Those are quite good figures. Because:
NimH AA at about C/10 (eg 230 mA for a 2300 mAh cell) have a fully charged Vf of 1.45V. This varies somewhat with temperature and manufacturer but not vastly. As C rate rises V rises. Charts are available. IF you are aiming at C/10 per cell use 1.45V/cell for fine design and 1.5V for rough basic design.   

1N5817 at 200 mA drops about 0.35V at 25C - falling with increasing temperature. See fig 13 in datasheet
Even at 600 mA (3 x PV panels) Vf typical is under 0.4V (fig13) .

Assume 0.4V diode Vf for design.
So battery needs 1.45 + 0.4 = 1.85V to fully charge. If wiring and connections etc drop 0.15V at 600 mA then 2V is fine.  In practice a bit more doesn't hurt SO their 2.0 to 2.5V  is about right. Note that the V specified is USUALLY loaded voltage in full sun. It is "not unknown" for small Chinese panels to be somewhat on the low side of their specification. The max current is at midday with the panel pointed at the sun. Output drops essentially linearly with light level. Voltage drops off much more slowly with decreasing light. 

Add another battery (cell) in series and you need another 1.5V giving Vpv ~= 3.5 - 4V = again jut what they say. !!!  3 batteries = 5V - 5.5V.
So a 6V nominal panel is a good choice for 3 x NimH. This allows it to still be useful in somewhat lower light conditions.


  My questions are: 1) What voltage of solar panel do I need? 


As above


  I did some research online and I can only find 2.5V panels from China


Almost all small panels are from China.
There are many many many brands and many many voltages and current levels available. 


  Do I need a charge controller? If so, why? According to this link: http://www.solar-electric.com/solar-charge-controller-basics.html/, if the panel outputs 2w or less, I wouldn't need a charge controller. 


Web advice on battery charging is often poor.
Older NimH allowed trickle charging at  &lt;= C/10
Modern higher capacity NimH allow NO trickle charging. For reasons why see other SE answers.
DO NOT trickle charge a fully charged modern NimH battery.
The best controller for solar charged NimH is a voltage limiter that limits the voltage that the per-cell voltage can rise to, to 1.45V/cell. Ideally this is applied per cell, but for not too many cells in series it can be applied to a number of cells in series. This can either be a voltage clamp which dissipates all PV energy once battery voltage is 1.45V/cell, or a series switch whih cuts off feed to the battery when Vbattery is high enough. In the latter case the battery voltage will change when Vcharge is removed and this needs to be accounted for with hysteresis. A good and cheap means of sensing battery voltage is to use a TLV431 reference zener, but this is by no means the only method. The TLV431 can be used to drive a voltage clamp or series switch. 


  3) If I use a 2.5V, 200mA max panel with a 1.2V, 6900mAh battery pack, I would plan on using 3 of these panels in parallel for a max output of 600mA. Does this seem reasonable? Or should I use more panels? 600mA would be roughly 10% of the battery capacity, which I have read you shouldn't exceed.


3 batteries in parallel give 6900 mAh as you say.
C/10 = 690 mA = OK.
NimH may be charged at up to C/1 (2300 mA for 2300 mAh cell) with no problems provided a robust charge termination system is used. This is difficult (or worse) with solar chargers as voltage and available energy may vary at any time and temperatures tend to be high and unpredictable. These factors greatly disturb standard algorithms.