I believe it make sense to break it down to separate questions, because each person may have answer to one or two of them of top of his head, but not other.
First of all, there is a well known MDM protocol which allow you to manage iOS device. If you can a documentation for it out of your Enterprise Developer program or some pieces of it out of reverse engineered document - https://media.blackhat.com/bh-us-11/Schuetz/BH_US_11_Schuetz_InsideAppleMDM_WP.pdf
However, you should be aware this is a protocol between a client and a server (meaning that it's not an administrative app, but rather administrative server which will send commands to a client)
Answering some of your questions:
2) Wipe, Lock and disabling camera exist in this MDM protocol. I am not sure what does it mean fixing WiFi and BT problems (most likely you won't find it in such a protocol)
3) This protocol allows to get a list of installed applications on a device.
4) Most of MDM providers have MDM server which does 90% of functionality of them and the rest is done using some mix of app wrapping, network compliance, interception of other protocols and so on.
5) No.
Additional to these. I think you need to break apart questions like: "How can I do X on a jailbroken iOS device". This way you will be able to build an app which will provide some additional functionality on top of MDM protocol.