At the powers involved you might melt quite a bit of the ground :). But some heating plants for centralized house heating do have electric water heaters, which are used, as soon as the electricity spot price goes below a certain number - the rest of the time they heat with gas.

These spikes would go mostly away, if more coal plants get replaced by gas plants which can be switched off very quickly. I guess, as soon as there are more of those days, this shift is getting financially attractive.

I doubt the power producers care what happens to the power since they just need to get rid of it. Given the rarity of these events and the cost of building something that can dissipate large amounts of energy very quickly I don't think it would be worth building infrastructure to do so. If it were, the power producers could just build their own.

Electricity doesn't work that way. To a first approximation, shorting something out actually uses no power, as there is no resistance/voltage drop. In reality, there is some resistance which takes the energy, and thus dissipates heat (in the scenario you're envisioning, likely the wires for a short time). Dissipating that heat is going to be your problem to overcome.