What type of energy should I use when describing the amount of energy available to increase entropy of a system?

The Helmholtz free energy for a system of N harmonic oscillators

How to derive the formula for heat produced due to electricity correctly from Joule's laws for heating?
From Joule's laws, we get this:
$H\propto I^{2}Rt$
$⟹H=K{I}^{2}Rt...(i)$
Now, we have to find/define the value of K. According to my book, when $1A$ of current passes through a conductor of $1\mathrm{\Omega }$ for $1s$, $1J$ heat is produced. If that's the case, then from $(i)$ we get this:
$1=K\times 1\times 1\times 1$
$⟹K=1$
Therefore, we get our nice little formula:
$H=K{I}^{2}Rt$
My question with the above derivation is, how did we find that when $1A$ of current passes through a conductor of $1\mathrm{\Omega }$ for $1s$, $1J$ heat is produced? Through experimentation? What was the name of the experiment and who conducted it?

Calorimetry - Emitted Joules
How can one calculate the total amount of emitted joules from an object with a temperature that isn't constant? A great start is this formula:
P = σ • A • (T^4)
If the formula is translated from sybmols to units it will look like this:
J/s = σ • (m^2) • (K^4)
(J/s) is joules emitted per second from an object. ( σ ) is the Stefan Boltzmann constant, 5.67•(10^−8). (m^2) is the area of the object. (K^4) is the temperature of the object, in Kelvin, to the power of 4. Now, if I transform the formula:
J = σ • (m^2) • (K^4) • s
Now one can get total amount of joules emitted ( J ) during a certain time ( s ), if one know the temperature and area. To the tricky part: what if the temperature isn't constant? The temperature will depend on how many joules that has already been emitted. And the joules that are being emitted will depend on the temperature.
How can I solve this? Do I need to combine this with an other formula?
Best regards! Please comment if something is unclear!

What is MIP hypothesis?
While reading about muon analysis I read that finding calorimetry segments along with the tracks in the silicon tracker for muons helps us find a subset of tracks compatible with the MIP hypothesis. I was looking for what MIP means and all I could find is Minimum Ionizing Particle. Does MIP mean that and what is the MIP hypothesis?

What is degrees Fahrenheit between the freezing and boiling points of water?

Calculate the entropy for 2.3 moles of neon gas at T= 289 K and p = 0.8 atm. State your answer in J K as a number in normal form accurate to 1 decimal place (X.X or XX.X, etc.). Do not include units in your answer. Calculate the Helmholtz free energy for 1.3 moles of neon gas at T= 339 K and p = 0.9 atm

The normal boiling point of water is 100 Celcius and the molar enthalpy of vaporization of water is 40.7 KJ/mol. What is the boiling point of water under a pressure of 0.500 atm?