Convert power + BNC CCTV camera to use PoE (power over ethernet) and have it able to obtain IP addresses

Solution 1:

However is it possible to plug the power + BNC(video) from the camera into the adapter, convert to PoE then take that newly adapted PoE line and plug that stright into a router / switch, instead of having to re-convert back to the original power + BNC(video) on the other end?

No. These adapters only convert wiring, but do not actually convert what's being sent over the wire – if it was analog video over BNC cable, it is still the same analog video over the Ethernet cable. (That's why they're "passive".) So you cannot connect it to an Ethernet switch, because the actual signal going over the cable is very much not Ethernet.

The other question is, even if this worked, would I be able to obtain an IP address from the camera or would this depend on if the camera had built in networking hardware?

Not with this kind of adapter, no, if the camera doesn't do networking, then the camera doesn't do networking – it won't suddenly learn how to speak TCP/IP over an analog-video BNC cable.

However, there do exist active converters which get their own IP address, take analog video from a camera, and send digital video over IP/Ethernet (just like a tiny DVR). They seem to exist on Amazon (I googled for "analog camera to IP"), and they're a bit larger and cost more than your passive wiring adapter. Note that in this case, it's the converter that participates in the network – the camera itself still has no clue about it.