This case would fit almost perfectly, and is watertight and sufficiently robust https://www.amazon.com/Plano-Guide-3600-Polycarbonate-Field/dp/B003FYMVXM/ref=sr_1_3?dchild=1&...-3 The case the driver set comes in is 216 x 175 x 35 mm, and this Plano one is 280 x 178 x 101 mm It is slightly oversized for the kit, but you could easily remedy that by placing or gluing in pieces of foam to fill the gaps.

Yes, it will need to be programmed with the VIN and the odometer reading

The starter relay in your fuse box could be welded. I would try removing it and seeing if it stops. If it doesn’t stop, there is a short to positive in the wire going down to the starter solenoid. If it does stop, I would try installing a different relay; often the horn relay or some other relay will be the same as the starter relay. If when you put in a known good relay, the starter still turns when you aren’t trying to start it, we’ll know it’s a problem with the control circuit to the relay, most likely a key switch.

Are you sure it is water? Give it a smell, if it smells sweet, it is most likely coolant, which can leak into the passenger side footwell because the heater core is mounted in the dashboard on that side. If it is a leaking heater core, it is a pretty involved repair that involves removing large amounts of the dash. If it is, in fact, water, I would go to a self serve car wash and try to look under the dashboard while someone sprays down the passenger side of the windshield, hood, and door.