Why sometimes my character at the beginning of a round draws less cards? (for instance, has only 3 cards at the beginning of a turn)
The standard is that you draw two cards from your deck plus one racial Move card. Thus, if you used all your cards on the last round, you should in fact have only three at the start of the next. Exceptions: On the very first round, you (and your enemies) draw more than this. If you have an active attachment like Accelerated Thought, your draw is modified as shown on the card.
You can also keep up to two cards at the end of a turn. So it may seem like some characters got 5 cards for the next round while others got 3, but the one that got 5 was actually holding onto the same two cards while drawing 3 new ones.