Before I get to the questions, let’s look at the battery issue. Reading between the lines, I’m assuming you actually want a 1.2 V power supply, not a 2.4 V supply. If that’s the case, then yes, wiring your 3 AAs in parallel will work and increase the capacity. That said, I’m curious as to what you are powering off of 1.2 V. Any Arduino-based system would require at least 3.3 V and likely 5 V.
As for the parallel/series combination idea, I would not use the configuration you suggested to get 2.4 V. It would produce 2.4 V when first plugged in, but the batteries might charge/discharge at different rates. Series/parallel combinations almost always comprise one series in parallel with an identical series. Laptop batteries, for instance, often combine cells in series to get the proper voltage, then put two or more identical series in parallel to up the capacity. In your case, doing that would require four batteries (a 2-cell series in parallel with another 2-cell series), and please make sure you use identical batteries and change them all at the same time.
An alternative is to make your power infrastructure (batteries, solar panel) at a higher voltage (say, 3.3V or 5V) and use regulators for powering your devices. This, of course, can be inefficient (less so if you can find appropriate switching regulators), but you might make up for that with the wider availability of off-the-shelf products at standard voltages.
Now, your questions:
(1) Yes, but it’s not an exact science. You want your panel to be rated slightly higher than your battery voltage to account for overhead and so that you can still charge in less than perfect, cool, sunny conditions. A 1.8 V panel would still charge your 1.2 V battery pack under sunny, cool conditions. a 3 V panel would charge under less ideal conditions but might necessitate some circuit protection if you are powering delicate electronics.
(2) Probably not. A charge controller essentially regulates the panel power into the battery and prevents the battery from being over-charged. Charge controllers are usually used in bigger setups involving high-capacity, 12 V marine (deep-cycle) batteries. For low-voltage/low current applications without delicate electronics (think solar path lighting), it’s not necessary at all. You might want to look up basic voltage protection for your data logger, but the batteries will probably be okay without a charge controller circuit. (Note: I’m sure there are experts on this topic on this forum, and I would welcome any additional comment from experience)
(3) That seems reasonable as a starting point. Remember, you do not need to charge from zero to full capacity every time the sun is out. The goal is just to have more power going in than going out on average. If you find that it dies in full sun, add solar panels in parallel. If you find that it works in full sun but dies after a few cloudy days, add batteries in parallel.