The octave will be no problem as it follows the same step pattern (although not quite sure what you mean by random notes…). However, you are right to raise the issue for the fifth. I’ve never tried, nor for any other interval and it is probably high time someone did. Well, I was wondering what to do before the football…
Again, it’s worth reminding readers we are talking about on-scale transposition, which creates a new step pattern (or mode); not key-shift, which keeps the same.