Is the Jupiter-Sun system considered a binary system of some type? Since Jupiter is very massive, it is the only planet (in our solar system) that has a center of mass with the Sun that lies outside the volume of the Sun. (Source)

If Jupiter was a star, they would form a « binary star ».

If the Sun was a planet, they would form a « double planet ».

Since the Sun is a star and Jupiter is a planet, does this have a particular name?

Does Jupiter have a special status or a particular effect in our solar system because of its heavy mass?

Since Jupiter-Sun's center of mass lies outside the volume of the Sun, that means that the Sun moves around that center of mass. Does this have an effect on Mercury, Venus, Earth and Mars orbits?
I'm not sure I understand your question entirely, but i'll do my best to offer a decent answer. It's true that the composition of Jupiter is very similar to that of the Sun (very similar approx. $H$ and $He$ abundance and pretty similar in density). The problem is that Jupiter is not nearly massive enough to have the internal pressure and temperature to undergo nuclear fusion. Jupiter doesn't have any particular special status aside from being the King of Planets in our solar system.

As for the last part of your question, all objects orbit around a center of mass. Though because the Sun is much more massive, the center of mass lies very close to the center of the star (except in the case of Jupiter, where the CoM lies outside of the sun and is approx the length of its radius). This is why all planets in our solar system orbit around the sun. This will indeed cause a slight perturbation of orbital alignment, but I don't believe it's significant.