What is needed to switch a relay via logic output (e.g. from Raspberry Pi)?
1) You cannot control the relay directly from an Rpi board, you have to use at least a transistor with base resistor (or a mosfet) and a flyback diode. That is what is on that small board.

2) no that is not how you should connect it. It will not work like that.

3) Yes there are other ways to control a different low voltage but I do advise to stick to a relay solution using that small board as it is cheap and reliable and you do not need to think about shared ground connections etc. You could implement the circuit from that small relay board yourself but believe me, it is not worth the trouble.