#1 - That's not a typo, it really is hc^2, not hc. There is a difference of a factor 4/c when you talk about energy or power, but here we are worried about intensity. The rest really is just math ...
#2 - Still just math ... but note that the constant 'sigma' you find here is not the same as the Stefan-Boltzman constant of number 7. Again there is a factor 4/c to convert between energy and intensity.
#3 - Just the Wien displacement formula, nothing more.
#4 - We'll talk about this in Thursday's lecture if you're having problems.
#5 - Using a fixed wavelength of 641nm, just calculate I(T) for both temperatures, using the formula from #1 and take the ratio.
#6 - Circular motion + electric force gets you velocity. Acceleration on a circular path is determined by velocity and radius (given). Kinetic energy comes from velocity, and power is energy divided by time. I'll field questions on this in lecture, there are a few steps involved.
#7 - The rate of heat loss is just the power emitted as thermal radiation, which you know to be P = A(sigma)T^4, where A is area and sigma is the Stefan-Boltzman constant. Power loss is due to thermal emission by the human, but the human is also absorbing thermal power due to the temperature of the surroundings (over the same surface area). Power lost minus power absorbed is the net power loss, which is the rate of heat loss.