Why is heat not measured in $rm kg,K$?

The kinetic energy (in Joules), for each atom in a gas, is $frac{3kT}{2}$, where $k$ is Boltzmann's constant and $T$ is the temperature in Kelvin.

The number of atoms in each gram depends on the atomic mass; for example, 12 g of carbon has Avogadro's number ($6.02times 10^{23}$) atoms of carbon.

The total heat energy is then the kinetic energy times the number of atoms, but not necessarily the kinetic energy times the mass.

You could look into specific heat capacity, measured in $mathrm{J},mathrm{kg^{-1}},mathrm{K^{-1}} $, which is a measure of how much energy is needed to heat (in Joules) 1 kg of a substance by 1 kelvin; there is a formula similar to what you suspected:

$E=mcDeltaTheta$

where $E$ is the change in heat energy, $c$ is the specific heat capacity and $DeltaTheta$ is the change in temperature.

Why the specific heat capacity varies from substance to substance was asked here.

Energy is not temperature. The lack of kelvins in the definition of the joule is not weird when you consider that every material requires a different amount of joules to heat up by 1 kelvin, and during a phase change, there is no change in temperature (kelvins) at all even as you add or remove energy (joules). You can add a bunch of joules to a material, and not all of it will show up as a temperature change. If I have water at boiling at 100°C producing water vapour, neither adding nor removing joules changes the temperature of either liquid or vapour. It just changes the amount of liquid and vapour.

Temperature is not a measure of energy. It seems what you are doing is you are interpreting temperature as a kind of "energy per unit mass", but that is not the case - otherwise we'd just use that: energy per unit mass (joules per kilogram, say), and we would not need a separate unit (kelvin) for temperature. It's far from unrelated to energy, and indeed, the higher the temperature, the more energy the system will have, but it's far from a simple, straightforward relationship between the two as you posit. In particular, two objects at the same temperature and with the same mass may very well contain different amounts of energy.

It's hard to give a solid intuitive definition of just what temperature "is" by itself - perhaps the best one can think of it as is the basic governing parameter of thermodynamic phenomena. Moreover, much of thermodynamics concerns temperature differences than just absolute temperatures: the more difference in temperature between two objects, the more of their internal energy can be converted into mechanical work, and the more readily the object of higher temperature dumps energy into the one of lower temperature. Indeed, it is these properties that can be used to construct a definition of temperature empirically, because they let us create an ordered set (a mathematical structure where you can assert a claim that an object is "lesser" or "greater" or "precedes" or "succeeds" another in some way) of categories of objects-at-a-given-moment on this basis, by comparing their energy flows. But the point is, it itself is not energy.

Energy, incidentally, is somewhat easier to explain intuitively - energy can be thought of as a kind of "money" that must be "spent" by one system to cause changes in another system, while the total amount of this "money" in the universe is a constant. Anything you want to do - accelerate an object, heat or cool it (i.e. change the temperature), lift it up, etc. - all require some amount of energy to be expended (which may in some cases be negative, i.e. the process gives energy to you.).

The reason energy has the units you mention is because of how the SI system of units is constructed - it's not something innate about energy itself. Energy is defined in terms of mechanical quantities: one joule is the energy required to generate mechanical work equal to pushing an object with one newton of force over one meter of distance. Force, in turn, involves mass and acceleration (Newton's second law) - thus, the units of energy ultimately will use mass, distance, and time.

However, there's no reason we have to do things this way. It's just based on the usual exposition of physics, starting with mass and motion first under the tired old Newton's Three Laws premises. We could start physics on other grounds, and then we would not necessarily have the unit of energy defined in terms of mass, length, and time.

One more note - you ask about what makes energy a scalar quantity. Being a scalar quantity is a mathematical property. In simple terms, it means the quantity is directly a real number, perhaps with units, and not drawn from a vector space over the real numbers.