Yeah pretty much. I don't really know what you mean by "synchronize their own home networks" but an AP will send out Beacon Frames to tell everyone that it's there.
Yes, but keep in mind (anyone correct me if I'm wrong..) that you will only send out probe requests for SSID's that your computer knows (ie, has connected to before)
Your AP will send out a probe response to ANY probe request. However, I would think that it would only send them to your neighbour's machines if they had either connected to your AP previously, or were currently trying to connect... I could be wrong.
Not sure to be honest. I think your machine might still send out probe requests to your AP, because it could be looking for a stronger signal from the same SSID in your ESS (another AP in your network with the same name. Mainly used for enterprise networks) and thus your AP would respond with a probe response...
Your machine will send out probe requests for any network it has connected to previously. If an AP is not sending out Beacon Frames (Hidden SSID), it will still respond to your probe request and you will connect, provided you have the correct credentials.
Hope that helps.
[EDIT]Actually, an AP with hidden SSID might still send out Beacon Frames, just not the SSID in that frame... I can't remember. It's been a while.[/EDIT]