Must a star belong to a galaxy, or could it be completely isolated?
In case it can be isolated (not belong to a galaxy), could it have a planet orbiting around it?
No, stars do not need to be inside a galaxy. It is estimated that about 10% of stars do not belong to a galaxy . While most intergalactic stars formed inside a galaxy and were ejected by gravitational interactions, stars can form outside of galaxies as well .
We assume that such stars could have planets, just like stars in a galaxy, although no specific examples have been detected yet.
 "Detection of intergalactic red-giant-branch stars in the Virgo cluster", Ferguson et al. Nature 391.6666 (1998): 461.
 "Polychromatic view of intergalactic star formation in NGC 5291", M. Boquien et al. A&A, 467 1 (2007) 93-106.
They probably have to be formed in a galaxy. But they can escape. There are many rogue planets that are found outside of solar systems. NASA has observed many rogue stars as well. This usually happens when galaxies collide and they throw out some of the stars. And yes, the rogue stars can have planets orbiting them