Does voltage produce a physical field? Does voltage produce an electric field, like how a current produces a magnetic field?
 A: If you have some charge $Q$, that charge creates an electric field.
The presence of electric field means that it takes work to push another charge toward (or away from) $Q$.
The voltage difference between two points $a$ and $b$ is defined as the energy needed to push a test particle from $a$ to $b$ divided by the charge of that test particle.
So, I wouldn't quite say that a voltage creates a field as much as I would say that when you have an electric field you have a voltage by definition.
In a more mathematical language, voltage is defined as 
$$V(b) - V(a) \equiv - \int_a^b \vec{E} \cdot d\vec{l} \, .$$
For a contant electric field this is just $V(b) - V(a) = -E d$ where $d$ is the distance between $a$ and $b$.
While current and voltage are the two "basic" quantities in circuit analysis, in pure electromagnetism the truly fundamental thing is, in a sense, charge.
Charge produces electric field, and therefore also produces voltage differences.
Moving charge, a.k.a current, produces magnetic field.
A: Having a difference in voltage between two points in space creates an electric field. You can visualize it like electron anxiety. the field is created by very anxious and claustrophobic electrons. It can demonstrate the path the electrons(or positive charge carriers, whichever you prefer) would take if released. They want to go to a place with lower anxiety(potential). 
