Yea that makes more sense, altho itachi had it for a longer time..Well its said that sasuke is better at using amaterasu than Itachi while Itachi is better at tsukoyomi so it makes sense.
Hakus not the only one with ice release, im pretty sure it was his clans kekkei genkai whereas with hashirama its his kekkei genkai not his clans.
And yea its haku's clan kekkei genkai, and haku was the only one who survived when his clan got killed/watever.