coloring outcomes from different coat colors in parents Questions

What is the likely outcome for coat coloring if an ESS sire is dark liver and white and the dam is black and white?
Do you “have” to have a dam with tricolor to produce? Do both parents have to be tricolor?

It all depends what the parents are carrying so you'd need to provide more info on their genetics

If dam doesn't carry anything then the litter will be black & white

If dam carries liver but not tri then the litter can be liver & white and black & white

If dam carries tri but not liver then the litter can be black & white and tri

If dam carries liver and tri, then the litter can be tri, liver tri, black & white and liver & white