Can I even use it to remote control my server? HP have iLO, but it's completely different beast, AFAIK. I've never heard about using Intel ME for servers.
ME is where the IPMI implementation lies. It can be used over the network to monitor sensors, to access the serial port and so on. It also has an HTTP server providing a web interface to the same functionality.
This is false. IPMI is a completely different thing from the Intel ME. It is implemented on a separate chip (bmc) and has its own NIC (usually with its own physical port)
AMT on the other hand provides its network functionality by hijacking the host's IP, and siphoning off traffic destined for its ports (16992 and 16993). This is one of the things that makes it scary, and just one of the differences between it and IPMI.
Source: I have worked extensively with AMT and built a semi-open-source Python management library for it in a previous job.
Are you sure of that? One most of the systems I've used with IPMI functionality (oracle and supermicro), the BMC resides in a separate chip with it's own processor and RAM.