In general terms, the URLs are case sensitive. In reality, some part of the URL are not case sensitive, but I think it's easier just to think that it is case sensitive.
As far as the /0 at the end...this indicates a particular layer. As feature services can have multiple layers in them, you can specify an index to get a particular one. If you don't specify the index, you get them all added to the web map.
thanks,
Mike