Can electric current be made to flow from + to - and - to + at the same time? And does it normally flow from + to - or - to +?
First of all, current doesn't flow. Charge flows. The conventional direction of the current is from + to -, and this direction can be the same as the direction of actual charge flow or the opposite, depending on the type of the charge carrier. As you know, the formula of definition for the electric current is I = dq/dt. Now, if we consider a conductor, in which the charge carrier is the electron which is a negative charge, the dq term will also be negative and so will be I in the direction of the electron flow. That's why the direction in which I will be positive is the opposite direction of actual electron flow, meaning that the electrons actually flow from - to +. If we consider a semiconductor, in which there are two charge carriers - the electron and the hole (which is a positive charge) - the electric current will be given by the flow of both (electrons and holes). While the direction of the actual electrons flow will be from - to +, the direction of actual holes flow will be from + to - (the same as the conventional current direction) because for the hole current the dq term is positive. So, yes, charge can flow from + to - and from - to + in the same time (for example in a PN junction) but the direction of the electric current is from + to -.
When Benjamin Franklin did his first experiments with electricity, he deduced that there are 2 charges and called them + and - He also discovered that only one of these charges can move while the other is stationary. He didn't know which so he made a guess and said the current flow from + to - which is wrong. Then in 1900, J.J. Thompson discovered an electron and proved electrons are negative and that they flow to a positive charge. In reality, - flow to + Physicists use: - flow to + Engineeers use: + flow to - Why engineers use a wrong assumption? Becuase during 1900's they didn't change their textbooks. So we are stuck with a wrong convection current.
I don't know the historical reasons for this convention, but I see this as a direct consequence of: 1) the definition of the intensity of the electric conduction current as I = dq/dt 2) the convention that the electron has a negative charge If either the definition of I was I = -(dq/dt) or the electron was considered to have a positive charge, then the conventional direction would have been the same as the direction of the electron flow. But I don't consider this convention to be such a twisted thing because electric current can also result from the flow of positive charged particles (positive ions in electrolytes, and holes in semiconductors), so electric current doesn't necessary have to mean electron flow.