There's several syslog options from AMP Setup -> External Logging.
You can configure AMP to send traps to an external NMS by setting up the NMS on AMP Setup -> NMS tab; and then creating System -> Triggers that have the external NMS set as a send alert destination.
Currently, the APIs require you to pass valid credentials into the AirWave system, so it depends on if you're authenticating into AirWave using the local users db (AMP Setup -> Users/Roles); or if you're using remote authentications (AMP Setup -> Authentication). This enforces that the data being fetched is the data that the user's role allows them to see.
That said, for up/down device counts, you're looking at the amp_stats.xml. If you get a response - that means the AMP is up. Otherwise you're looking at getting the actual up/down state for an AMP from a Failover AMP's trigger/alert. The APIs do not collect AMP configs or push AMP configs - that'd be a feature request.
The question points you have are somewhat general, so having a better idea of your use case could help. The APIs are specific to the data that the AMP contains, so unless you have Master Console or Glass in play - you'll have to collect individually from each AMP.