Is it possible to create a differential if I don't have it as a ready to use element? I have numerous gears, axis and other elements from different Lego Technic sets.
I wonder if using them I can make a kind of a differential like this?



If not, how can I implement similar functionality in case I have two motorized wheels on one axis that can be steered and connected to motor(s) via cardans like this:
This can definitely be accomplished, although it will be difficult to create a differential as small as the dedicated piece that you mentioned.

Some builders create their own differentials for use in larger models because of the inherent torque limits and slippage that the traditional differential and single bevel gears introduce. Here's one example (source) of a differential that can be created using fairly common elements:



The key is to build the frame containing the three gears and then attach a fixed gear to the frame. The tricky part is finding a gear that can pass an axle through the center without being fixed on it. In the example above, half of a turntable was used, but there are a few other parts that could work as well. Here's the turntable (48452):



If you are comfortable with modifying your parts, you could also drill out the center of a gear to allow it to turn freely.