So let me try and answer your questions. Advertising is when bluz is sending out the standard BLE advertisement packet, which broadcasts the name and other data about the device. The advertising interval is strictly how often these packets are sent out. A central, like our gateway, detects these advertising packets and determines if it should connect or not.
Once a device is connected, the connection parameters are used to determine how often bluz and the central should sync up. So they will chose a connection interval, which just means that the central and bluz must talk to each other that often. This is also when data can be sent. So if the connection interval were 20 milliseconds, then bluz will transmit a data packet every 20 milliseconds to the gateway, and the rest of the time the radio is off.
The connection interval is chosen when the devices connect, and ultimately the central choses it, but it will ask the peripheral what it would like to support. So setting it on the central guarantees the right interval is chosen, but setting it on bluz may work as well with some central devices.
Once connected, bluz will stop advertising. So disconnecting and stopping advertising are two distinct actions.
Yes, waiting for a connection and then sending data will work fine. I hope that answers all your questions, but please let me know if you have more.