Is it possible for a red dwarf to orbit a gas giant?
Has this happened and it is just assumed that the gas giant is orbiting the star?
Pretty much by definition, no.
In order for "object A" to orbit "object B", "object B" needs to be substantially more massive than "object A". In order for your red dwarf to orbit something, that "something" needs to be more massive than a red dwarf. With the current composition of the universe, that "something" will be mostly hydrogen, and once you reach a certain level of mass (about 90 times that of Jupiter), you can't prevent nuclear fusion from starting. The definition of "planet" is a bit fuzzy, but everyone agrees something lit by nuclear fusion is a star, not a planet.
This could theoretically change in the distant future, when there's enough non-hydrogen mass around for stellar-scale rocky bodies.
Neither statement is true, just like the statement that the "Earth orbits the Sun" is not strictly speaking true.
In truth, if we are talking about a system with two masses, what happens is that they both orbit their common centre of gravity. Whether we perceive that one object orbits another is really just a question of their mass ratio. In the example shown below, the smaller, less massive object might be said to orbit the larger object and follows the red path. However, the larger object also executes a smaller circular orbit shown in green, with the same orbital period. Both objects actually orbit the centre of the picture, which is the centre of gravity of the system.
In many cases - e.g. a planet like Earth and a star like the Sun or a satellite around the Earth, the mass ratio is so small, that it does make sense to make the approximation that one object orbits another.
Well, technically, a planet and a star orbit each other. In that sense, yes. If a gas giant is orbiting a red dwarf, then the red dwarf is also orbiting the gas giant. Always.
If you quibble that it doesn't count if the star doesn't move much, I throw the quibble back by pointing to Jupiter and Sol. The barycenter of the system is outside the sun, and that means the sun's orbit is not exactly tiny. (Smaller than Jupiter's orbit, but now we're no longer talking absolutes.)
I can see how it might be possible for a gas giant to out-mass a red dwarf. The gas giant could be really a protostar, on its way to becoming a red dwarf, but not there yet. That would mean the gas giant would have to be a lot younger than the star, meaning that they formed separately and one of them captured the other.
Alternatively, the red dwarf could be a gas giant that got heated by some external source, say by a nearby normal star. -- No, scratch that. External heating would keep it from contracting, and its core would never become hot and dense enough to initiate fusion. -- Unless the red dwarf was a gas giant that swallowed a neutron star or black hole.
All pretty unlikely, but not downright impossible. And in astronomy, whatever is not prohibited is mandatory. It it can happen, it has.
Simply said if a star would orbit a planet, the planet would become the star and viceversa. Hm... never seen that. But who knows. The universe is so vast and the physics we know only covers an insignificant percent of all phenomena in the universe. You just imagined that, so in another universe this is true. It is your universe!