You want large area of the ice to improve heat transfer. And you want good thermal contact between the ice and the contents of the box.
For this reason, crushing the ice and leaving the water in would be good. Gently stirring the contents would be the finishing touch (heat transfer by convection is much more efficient than by conduction, and the work done in stirring will be quite small compared to the benefits).
As for the other points: wrapping in foil limits convection -> bad. Single block of ice limits area -> bad. Draining the water means that the heat capacity of the convection medium (air instead of water) is lower -> less ability to carry heat from A to B, bad.
The point that crushing the ice does work is true but irrelevant. The question was about rate of cooling, not maximum final amount of heat extracted. Assuming that the contents of the box will end up at 0°C with some ice still solid, the work done will not materially change the outcome.
By shifting the melting point to a colder temperature, the surface of your salt water/ice will be below $0C$. Assuming the environment is warmer than that, the temperature difference will be greater with the salt water ice bottle than the pure water ice. This implies greater heat transfer and faster melting.
If you want it to melt more slowly, the simple answer would be to put a layer of insulation around the bottle. The rate of melting is related to the rate of heat transfer. This means your desire to melt as slowly as possible is equivalent to wanting it to cool the environment as slowly as possible. Insulation will accomplish this.
You can think of your bottles as having almost a set amount of cooling power, rather like the amount of energy in a battery. Adding salt doesn't increase the amount of cooling power. You can slow down how fast the battery is used, but that doesn't make it more useful. Here you can slow down how fast the ice melts, but that might not make it cool your aquarium more efficiently.
Best Answer
First, I think there's some interesting physics to that question and I don't think the dry ice bomb comparison works because a dry ice bomb is a sealed system. Pouring hot metal onto a block of ice shouldn't create a seal because any escaping hot water vapor should be able to fly out the hole that was created by the copper that was poured in, even as part of the copper solidifies the combination heat and liquid water around the hot metal and hole it made going in should allow gas to escape and never create a tight seal. That's true unless the block of ice is cold enough to re-seal after entry, but I don't think that's the case here.
What happens in that video is basically the same thing that happens when you drop icecubes into a warm drink and the ice-cubes crack. Ice's density drops with temperature. As ice grows colder it grows more dense and as it warms it expands. This is called differential expansion.
When that guy in the video poured molten copper into the block of ice, the same thing happened, only the expansion happened from the inside out, so ice was expanding from the inside, when the outside of the ice wasn't, and ice, not being very elastic, once the pressure reached a certain point - kaboom. It's logical to think expanding gas created the explosion, but I don't believe that's the case, I think it was the expanding solid ice from the inside which created the explosion, loud noise and flying bits of molten copper, and most important, 3 million views.
Graph here on how much ice expands by temperature.