I know that harmonic analysis can be done like this:
i iv V i
Is there a similar standard for melodic analysis? Is melodic analysis a thing?
Yes. There is a whole discipline for it: http://jan.ucc.nau.edu/krr2/mel_anal.html
One looks a range, key, ornaments, type of tone connections, essential and non-essential tones, rhythm, harmonic implications, etc.