The term virtual is used in other places in physics. For example in virtual images in a a mirror : we see an object in great verisimilitude, even ourselves. Why is the image called virtual and not real? Because it has the optical properties of the imaged object but not a large number of other attributes, mass being the simplest. In addition, its existence is not independent of the object, a prime attribute of reality macroscopically.
The term virtual coupled to the term particle comes from the shorthand of Feynman diagrams. Before Feynman invented his diagrams the study of elementary particle interactions was very complicated, involving convoluted integrals in series expansions .
electron electron scattering, time upward in y.
Feynman diagrams allowed to order the perturbation terms according to complexity, which led to seeing easily the more dominant parts of the expansion in series, ordered in "orders". This is a first order diagram and the calculation has within the integral a pole related to an exchange of a "particle" that has all the attributes of a photon, except its mass. It is off mass shell, i.e. the four vector that transfers energy and momentum from one electron to the other looks like a photon, the way the optical image of you looks like you, but is not an on mass shell photon, so it is called virtual.
Feynman diagrams can be very complicated, like these that allow the calculation of top production:
top production one of diagrams.
In this diagram for the calculation of the crossection for top production real , i.e. on mass shell, are the incoming on the left and the outgoing on the right particles. All the rest are virtual. Note the exchange of W which has a mass close to 100GeV on shell, it can have any mass consistent to the maths of the diagram, but it carries the quantum numbers of a W boson.
An electron travelling from one point to another will take all paths, emitting and reabsorbing virtual photons. Does the electrons take all paths including those that involve faster than light velocities?
Here you are mixing the Feynman diagram approach of calculating interaction crossections with the path integral approach where the term "virtual particle" has little meaning. In this calculational approach this article might help which explains simply about fields and the real particles being excitations of fields.
In the Feynman diagram approach one needs an interaction, the electron running along is not interacting with anything that can be framed as a Feynman diagram, therefore it has no meaning to say it is emitting and absorbing virtual photons let alone their velocity.
