How do I connect digital logic components (eg. AND gate) to parts in a circuit?
I would use perfboard. 


  (Perfboard) is a thin, rigid sheet with holes pre-drilled at standard intervals across a grid, usually a square grid of 2.54 mm (0.1 in) spacing. These holes are ringed by round or square copper pads


You can solder all your breadboard compatible components to that. You might need something to mount the lever to, though, depending on it's form factor

As for how to turn NAND gates into AND gates, you just need to invert the output of the NAND gate. You can create an inverter by tying both inputs of a second NAND gate together. Then you simply feed the output of the first gate to the input of the second.