"My understanding of the 2nd law of thermodynamics is, partly that, basically a 'cold' object cannot heat a 'hot' object."
That's only true in some situations. This is a problem of over-simplified explanations in physics education. A situation is described - spontaneous heat flow between otherwise-isolated bodies at different temperatures - and the second law says that in this situation the net heat flow is from the hot to the cold, so the hot body cools down and the cold body warms up. It's never in the opposite direction, net heat flow from cold to hot, cooling the cold body and warming the hot one. But it only applies to spontaneous heat flow, isolated bodies, and is only talking about the net flow.
The obvious counterexample is a refrigerator, that pumps heat from the cold bodies inside to the warm outside. That particular phrasing also proves hugely confusing in the study of black body radiation, because heat is transferred in both directions, both hot to cold, and cold to hot. The net flow is still hot to cold, but the loose wording misleads a lot of students into thinking the flow of black body radiation from cold to hot cannot happen.
"Case in point: if I take a 1500W hot plate and put it on full, the plate will get to a certain temperature, say 130°C."
There are two ways this can happen. You can hold the external power input fixed (1500 W) and work out the equilibrium temperature as it escapes. Or you can use a thermostat to turn the power on and off to hold the body at 130°C. What happens depends on which you are doing.
If you are using a thermostat, then the hot body stays at 130°C regardless. Put an insulating dome over the top, and instead of running full time at 1500 W the heater starts turning on and off, putting less power in. The air and the dome heat up. The outside of the dome radiates to the void, but can never reach 130°C because it's surface area is bigger than 1 m$^2$ and it's got a maximum of 1500 W input even with the heater running full blast. Since the heater isn't running all the time, the outer temperature is even lower.
This is the situation of a human covering themselves with a blanket. The body has a thermostat, and can supply about 100 W when at rest to keep itself warm. If the surroundings are cold enough that 100 W is insufficient, skin temperature starts to drop. A blanket resists and slows the heat flow. The body warms the inner side of the blanket, which radiates back and reduces the outflow. The body needs less heat to maintain its temperature. The outer side of the blanket is colder than the naked body, and radiates far less heat to the cold surroundings. Blankets mean people can maintain their body temperature with less power expenditure.
The other situation is where the heater generates 1500 W, no matter the temperature. In that case, the dome radiates more heat back, and the temperature of the heater rises far above 130°C. The temperature keeps rising until the outside of the dome is radiating 1500 W to the void. The outside of the dome will be at a bit less than 130°C, because it has a larger area. But the heater in the middle will be a lot hotter than 130°C, so that the net heat flow from heater to dome will remain at 1500 W.
A nice analogy for this situation is to have a hose pipe pouring water at a fixed rate into a leaky bucket. The water level rises until the same amount is pouring out of the holes as the hose pipe is supplying. If you put one leaky bucket inside another, you can get the water level higher, as the flow is determined by the difference in water height between the inside and outside of each successive bucket. If the bucket never leaks or spills, there is no limit to how high you can get the water. But you can never get water to spontaneously flow up hill, from a low water level to a higher one. You have to pump it.
Similarly, you can get even higher temperatures if you feed the power into a volume with a much smaller surface area. Consider an electric arc welder, where we feed 1500 W into a volume a few millimetres across. The temperature needed to blackbody radiate 1500 W through such a small area is considerably higher than 130°C! The power input on its own does not set a limit on temperature - it depends on how easy it is for the heat to escape.
Another good example is the sun. The heat generated by fusion in the sun's core is roughly the same per unit volume as a human body. But the sun is big! The total heat generated increases as the cube of radius, but the surface area only increases as the square. So big things tend to have more difficulty losing heat. The centre of the sun is so much hotter than its surface (10 million degrees compared to 6000 degrees) because the centre is thermally insulated by the mass surrounding it. If you cut out a human-sized chunk of the sun's core, somehow kept it compressed at the same pressure, but able to radiate freely to space, it's temperature would soon be the same as that of a human in the same circumstances. It's the incredibly thick 'dome' of outer material radiating heat back that keeps it so hot.
