About the pressure sensor: I think Putyn is right about the sensor in an enclosed case.
As well as reading the pressure inside a pressure sealed box your requirements pose a few other challenges.
Seeing the warning LED through the aluminium is going to be tricky. Reading the temperature through the metal should be possible but the reading will be delayed. Getting the processor which is off to detect when the dive starts and turn itself on seems to need time travel (okay you could have some electronic circuit that switches the processor on when electrodes get wet, but that's making things complicated). Reading the surface atmospheric pressure, to adjust for altitude, after the dive has started also needs a bit of time travel.
I think life would be a lot simpler if you just turned the thing on/off manually.
Take a look at your depth gauge. I imagine it is liquid filled and has a transparent face which, when pressed, causes the pressure reading to change. You might want to consider if you could do something similar putting the arduino inside a container filled with inert non-conducting liquid, flexible transparent front to transmit pressure and allow led to be seen, thin metalic foil transmitting heat from exterior to temperature sensor surface.
Some dive watches are liquid filled. I bet that some dive computers are too.