Critical Strike will work with any weapon. Critical Strike is a very important Amazon skill because it adds the chance of doing double damage. Since it's passive it can combine with any of the Amazon's skills. The beauty of Critical Strike is that it is a Passive skill so it is always "active" and it will combine with additional skills you may use such as Multiple Shot, Pierce and so on.
Does the Amazon’s Critical Strike double all of the damage done, or only the base damage done without any double in elemental attacks?
The rules for what modifiers get applied when are very complex. In general, all bonus multipliers are added together and then applied (+50% and +20% = +70%) and then all bonus adds are then applied. Some magic modifiers affect the base damage of the weapon before the bonus multipliers - generally the non-elemental ones.