Powering Peltier using solar panels
Your first decision is to figure out the heat load on the space. How many BTU/hr you need to remove to keep the temperature you want. Then figure out how large / how many Peltier you need to remove that much heat. Do you need some reserve capacity for unusual circumstances or is a higher inside temperature Ok sometimes? Then you need a power budget - how much power will it take to remove that much heat with Peltier panels. How much PV area will you need to provide that power? If you need cooling when the sun isn't out, your power budget has to include the 24-hour demand, and the intermittent supply, perhaps again with some headroom. Power storage - batteries or the power grid - will be a requirement to match the two. The same storage can power your Arduino. The Arduino's power budget will be minuscule compared to the rest of the installation and you can totally ignore it. Do you have to power your internal network that connects the Arduino to the Internet, too? Figure that in. By this point, you'll have the answers to most of your questions. Just a heads-up: A Peltier beer-cooler is quite doable. A whole house A/C might be doable but no practical - you probably won't like the numbers! I'm guessing your project lies somewhere between the two.