I'm sure you could plug it into a (cable/DSL/whatever) modem equally well.The Pi can connect to a router by ethernet cable or Wifi
Only if the device on the other end of the cable supports it. Regular ethernet doesn't provide power, you need PoE (Power over Ethernet) kit at both ends for that.
If you have a 3B+ then the PoE device must be one that supports Gb. A 10/100 PoE injector / Splitter won't work.
It cannot be powered by an Ethernet cable directly. There are POE (Power Over Ethernet) systems that could work, but that's much more involved (not to mention much more expensive) than just using the official power supply.
Yes, I have tested my Pi computers with cards as large as 128GB, but they should be compatible with cards up to 2TB (which don't exist at this time).
Yes, that can be done.
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