This raises a question for me. When and where does the power company convert DC to AC ?
Usually, never. Almost all power is generated from rotating machines that naturally produce AC (solar photovoltaic is the exception, being naturally DC and usually converted to AC close to the panels).
Power companies do nowadays use some DC links for special purposes (notably between independent grids), but that's not relevant to charging.
Is it all possible to eventually have DC charging at home ? What would be involved with this ?
There's nothing magic about DC charging. In a sense, all battery charging is always DC (at the battery terminals). All we mean by DC charging is that the charger is moved outside the car - if the charger is inside the car, we call it AC charging (because the plug has AC on it), and if the charger is outside the car we call it DC charging (because the plug connects direct to the battery and so has DC on it).
In the case of Tesla and Supercharging, that is literally all the difference there is: the superchargers are built from the same modules as the chargers in the cars. So if you removed the two chargers from your car and put them in a box outside you would have a low power supercharger. In that sense, you could easily have DC charging at home if there was any reason to do so; Tesla could easily build a mini-supercharger for home use if they thought there was a market for them.
I suspect the real question you are asking is based on the misconception that "DC" is the critical part of "DC charging", and so having a supply of DC from the power company would be helpful. In fact, as above, "DC charging" is just a shorthand for "charger outside the car". If the power company suddenly started delivering DC direct to your home (highly unlikely), or you had a solar system that meant you had a local supply of DC (quite plausible), this actually wouldn't help at all - you would still need a charger between your source of power (be it AC or DC) and the battery.
The conversion from AC to DC is really only an incidental function of the charger - its primary function is to regulate the current and voltage applied to the battery. In fact, the electronics in your existing 'AC' charger are almost exactly what you would need to make a charger that took DC at its input. These chargers invariably take whatever input they are given (usually 50/60Hz AC, could be DC) and convert it to very high frequency AC as part of the regulation process.
As an example, these people make a CHAdeMO charger; the standard model runs off normal AC, but they can also supply the same unit configured to take DC from solar panels, or DC from the battery of another EV.
http://www.andromedapower.com/Home_Page.html