How many turns does the move Salt Cure last?
Salt Cure doesn't doesn't have a turn limit. In this test replay, Salt Cure's effect kept lasting until Pelipper fainted. This implies that it works like the Ghost-type curse, Salt Cure's effect chips you every turn and doesn't have a turn limit.
And does in bind (i.e. prevent the target from switching out)? If not, then does it continue to deal damage to the new opponent if the affected opponent switches out?
Both of these questions are answered with no, as shown by this interaction in the replay I linked above:
Turn 12
Kartana used Sacred Sword!
It's not very effective...
(The opposing Garganacl lost 12% of its health!)
The opposing Garganacl used Salt Cure!
It's not very effective...
(Kartana lost 7.7% of its health!)
Kartana is being salt cured!
The opposing Garganacl restored a little HP using its Leftovers!
Kartana is hurt by Salt Cure!
Turn 13
Kartana used Sacred Sword!
It's not very effective...
(The opposing Garganacl lost 13% of its health!)
The opposing Garganacl used Recover!
The opposing Garganacl had its HP restored.
Kartana is hurt by Salt Cure!
Turn 14
Kartana, come back!
Go! Swampert!
The opposing Garganacl used Iron Defense!
The opposing Garganacl's Defense won't go any higher!
Kartana was able to switch out even though it got hit by Salt Cure. When Swampert got swapped in, Salt Cure's effect didn't deal damage to it. However, if the teammate gets swapped in and Garganacl uses Salt Cure on the same turn you switch in, the teammate will take chip damage from Salt Cure.
TL;DR Salt Cure's effect works like the Ghost-type Curse. Salt Cure's effect lasts until the Pokemon hit by it faints. You can switch out after being hit by Salt Cure, and Salt Cure's effect goes away when you switch out. That means Salt Cure's effect doesn't damage new teammates being swapped in unless Garganacl uses Salt Cure on the turn you switch in.
