Okay, so I'll try to cover everything in a list of bullet points:
1. You need to have taken damage to trigger Lily's scene, so auto-wins won't work (this will be changed in the next update). To find more fights, follow #2 below
2. To gain more Goons you'll need to either do more Goon fights (get Hana to make you "Caviar on Toast" to increase the chance that the Org will try to take back one of your territories) or get Hana to make you "Fresh Baked Bread", which will give you 5 troops per day.
3. The Nexus is unattainable for the moment. Work in Progress.
Hope that helps clarify some things!