I sent a Seek the Lightening Thief adventure yesterday and had it waiting for collection when Chapter 4 started. It DID count as my first one.
I can confirm Chapter 4 requires 2 adventures.
Also, the Skyfall Cerberus does NOT drop fur during goals 1 and 2. The Greek statue and marble DO drop as soon as Chapter 4 starts.
Chapter 5 requires neighbor requests before Seek The Lightening Thief, so I sent a 22 hour adventure and will send the LT adventure tomorrow.