If you have your diode module it should fit inside that heatsink and lock in with a set screw which is provided with that specific item.
The spring that sits between the diode and the lens provides tension so that the lens doesn't move around too much (closer or farther from the laser diode). The amount you screw the lens in is up to you because it determines how the laser is being focused.I believe the actual diodes themselves are just press fit inside a housing meant for the specific diode standard, maybe they add thermal glue? As for a larger outer heatsink they generally have a set screw that clamps the diode's housing in place.Those two have a combined voltage of 7.4v so that should work fine with the circuit.
Using the 8Pin ATTINY programming shield with an external clock
PWM and Frequency Controller (for lasers and diodes)