What would the consequences be of Henry III not being chosen as king of the PLC? Would La Rochelle be taken and if so how would that bear on the war with the Huguenots?
Probably not.
The siege of La Rochelle was not going well for royal forces.
If Henri d'Anjou stays in France, the Malcontents Plot as we know it may be avoided. However, the Malcontents would still be there and would rebel sooner or later as in OTL.
The main difference for France is that Henri d'Anjou probably never meets Louise de Lorraine-Vaudémont (he met her during his travel to Poland in OTL). As a result, Henri III marries someone else, probably Elisabet Vasa as Catherine de Medici wanted. That may give him a surviving son, avoiding the succession crisis and the Bourbon's rise to power.
What about Poland-Lithuania ? Who would be elected and what would be the consequences ?