Ceiling speaker has 3 wires

The rain collector appears to have only a reed switch contact closure output. The collector itself will not produce voltages on the wires. Instead you have to connect the reed switch contact wires into a circuit that puts a bias on the wires that can be detected by the Arduino.

Since the wires are most likely running from outdoors (wet rain location) to indoors (dry Arduino location) it is a good idea to prepare some additional protection circuitry before the input to the Arduino so that if the wires experience some electrical disturbance it does not knock out your MCU board. You could try the following circuit as a starting place to hook up the rain collector reed switch.

1: Yes, you can attach any number of LEDs (or pretty much anything else you like) to the relay, so long as you ensure that there is never more than 5A of current drawn through it. CAVEAT: The LED lights you linked didn't have a listed wattage or amperage. Without knowing how much power they draw, we can only guess as to whether or not the pair of them could take more than your relay's max 60W/5A.

2 & 3: Yes, and for vehicle wiring, soldering or crimp-on connectors are the preferred attachments as the vibrations can cause other connectors (like wire nuts) to work loose.

4: Yes, any wire rated for at least 5A/12V should be safe to use with that relay.

Image: Yep, that's how to do it.