Could an icy/extremely-cold asteroid/comet ever strike and cool the Earth?
Could an icy/extremely-cold asteroid/comet ever strike and cool the Earth?

No.
The smallest possible velocity is about 11 km/s velocity, and that would require a near miracle. For that to happen, the object would have to enter the vicinity of the Earth through a very small keyhole near the Sun-Earth L1 or L2 point with a very small velocity (a comet will not have that very small velocity), then be nearly ballistically captured by the Moon, and then fall to Earth. A comet would be more likely to hit the Earth at 40 km/s, or larger.
And that means that it does not matter how cold or how hot the object was prior to impact. The kinetic energy of the impact will dwarf whatever thermal energy the object had prior to the impact.