Is it possible for a star to orbit a planet? Is it possible for a red dwarf to orbit a gas giant?

OR

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.