If someone asks me to play I VI II IV in C minor, am I supposed to play them in the same octave? Like, if I start with C minor in the third octave, do I then move 6 steps up to play G# major or can I also move down 2 steps and play the same chord an octave below? Will it still be the same chord progression?
Yes they are all the same chord progression. When just given the Roman numeral analysis or chord symbols of the progression, the exact voicings do not matter nor does what octave you play them in.
However it should be noted that what voicings you use, what octave you play in, and how much you move from one chord to another affect how what you play sounds. Try different voicings in different octaves and see what you like best.
Whether you move up or down, the progression will sound the same. Which way you choose purely depends on the song. There are times were the composer (or some other musician) will specifically ask you to move a certain way; if not, try both of the ways yourself and see what fits best.
If the melody moves up, it might sound nice to move along with it. It will also sound interesting if you moved the opposite way. If the melody goes up, try going down and vice versa.
Also, it might be nice to mix those two ways. The first time go down and the second time go up.