Haunt does jump to another enemy when its current host dies.
Only one Haunt can be active on an enemy.
Normally Haunt does not target an affected enemy, it finds a new one. If there are no more unaffected targets, Haunt replaces one on an already affected one, refreshing its duration. When spamming Haunt, you usually do not bother about targeting or refreshing - everything will be affected very soon, especially with Resentful Spirits and Haunting Girdle.
All that said, it does not actually matter for the Jade Harvester build. With multiple spirits and spamming, everything will be affected in seconds. With Creeping Death, there is no worry about spirits expiring. Main damage comes from Soul Harvest detonations, which consume all Haunts anyway (if anything survives). 2-piece effect of double Haunts does do its part too, but it is mainly useful on stragglers, when dropping a Soul Harvest it is not worth it (thus there are hardly any unaffected targets around).
This is an easy and powerful build, go for it.