One of the things I'm looking at is the effect of heat upon starships, such as the correlation between starship armour values and heat resistance, the effects of added heat shielding, and the effect of tech levels on both.
The concept you are looking for is basically black-body radiation.
https://en.wikipedia.org/wiki/Black-body_radiation
Everything radiates heat. You gain heat from external and internal sources. External sources are e.g. a star shining on you. Internal sources are the waste heat from the power plant and any power using equipment onboard.
Heat is energy, energy cannot be created or destroyed only transformed. Each Joule of energy produced on board (transformed from matter into energy) will become some other form of energy such as kinetic energy (motion), but most of it will eventually be transformed into the most dispersed form of energy: heat.
Even small starships produce massive amounts of energy. A Scout might produce 500 MW or 500 million joule / second useful energy (and a lot of waste heat). Much of the useful energy will be used internally and transformed into heat. Totally the Scout might need to get rid of 1000 MW of heat continuously.
The only way to get rid of heat in space is to radiate it away using a radiator.
https://www.nasa.gov/mission_pages/station/structure/elements/radiators.html
In Earth orbit the Sun's radiation is about 1300 W/m². The inverse square law (that you have used correctly) says that at twice the distance the effect is one quarter, and that at half the distance the effect is four times bigger. Mercury has an orbit round the sun about a third of Earth's, so the radiation is about nine times bigger, lets call that approximately 10000 W/m² for simplicity.
A Scout might have 37 m × 24 m / 2 ≈ 440 m² in the sunlight. It would absorb at most 440 m² × 10000 W/m² ≈ 4,4 MW as heat in Mercury orbit. But we already know that the Scout has to get rid of 1000 MW to be able to function. The added heat from the Sun is almost negligible.
Heat shielding will only slow down how fast you absorb the heat, not how much heat you absorb. It might be useful in a half hour landing operation, but not so much for weeks in space.
Another thing I was looking at is how to calculate heat from a star given distance and stellar temperature and/or luminosity.
https://en.wikipedia.org/wiki/Black-body_radiation
Scoll down to "Temperature relation between a planet and its star".