Not UA specific.
Two international trips:
Trip one: JFK to Geneva - Sept 1
MXP to JFK - Sept 25

Trip 2: JFK to Dubai - Dec 3
Dubai to BCN - Dec 16
BCN to JFK - Jan 6

Can I book Delta (open Jaw) JFK to GENEVA - Sept 1
BCN to JFK - Jan 6

Emirates MXP to JFK - Sept 25
Emirates (rewards) JFK to Dubai - Dec 3
Emirates (rewards) Dubai to BCN - Dec 16