1. Currently, you can create personal accounts (no fee) for non commercial use. Recently, we introduced ArcGIS Online subscriptions, which is fee based and provides additional features, such at the ability to create services in ArcGIS Online of your data.
2. Yes, currently there is no fee for this use, as long as it's non commercial.
3. If you created a web application and shared it with the group, I'm guessing that you didn't also share the web map with the group. It needs to be shared otherwise nobody else can see it.
Thanks,
Mike