Is it possible that dark matter and dark energy come from the same dark particle?

No. Dark matter is mass that can move through spacetime. It scales with the expansion of the universe $a$ as $a^{-3}$. In contrast, dark energy is a property of spacetime itself and scales not at all with the expansion of the universe (which is why it's also called a cosmological constant)

Not in the sense that you ask. In cosmology, matter is anything whose energy density scales with $a^{-3}$, where $a$ is the scale factor. Dark matter is called dark matter because it is an unknown component of the universe that nonetheless obeys this scaling relation. On the other hand, dark energy does not scale - it is proportional to $a^0$. See the technical definition section of the Wikipedia articles on dark matter and dark energy. Any particle would necessarily be dark matter but not dark energy, because the density of any particle in a box decreases as you increase the size of the box.

However, there are theories that attempt to explain dark matter & dark energy with a single framework. Again, see Wikipedia on dark fluid.