Intel ME has independent NIC access and usually cannot be disabled by the user.

It has only been six days since it has become possible to neutralize the ME firmware for Sandy Bridge and Ivy Bridge[1].

[1]: https://news.ycombinator.com/item?id=13056997

The ME is basically an independent universal computer in its own right, it comes with its own clock, RAM, CPU etc...

It is like a Matryoshka doll sitting inside the Intel CPU of your computer. Therefore, yes, it can contain all of that. For further details see my other two posts.

How else would you implement remote server management that works even when the hardware is shut down (put still connected to power) if not with a seperate CPU that has access to the NIC?