GUESS WHAT THIS IS:
∧ ∧
⊂∵ω∵⊃

a cat

Explanation:

c, d, e, f, g, a, b, and c.

sometimes western music will also use solfege. the names of these are do ra me fa sol la si (or ti) and do.

to indicate a note a half step higher or lower then the original note, you would say c # (for a half a step above c) or bb (for half a step below b)

i'm not sure i completely understand your question, but i will do my best.

i think the answer you are looking for is sequence. (see below)

a repeated melodic fragment would be best defined as repetition, while repeated chords (different or the same) could be a chord progression. however, you then mentioned the chord and melodic fragment beginning on higher or lower pitches, in that case, the answer could be best defined as a sequence. the definition of a sequence is as follows: "the restatement of a motif or longer melodic (or harmonic) passage at a higher or lower pitch in the same voice."

let me know if this !