Is it possible to check an Ethernet twisted pair with a multimeter?
The only thing you might be able to test is the continuity of the cable.  If you are attempting to test to see if data can traverse the cable than this is an okay way to test.  If you are looking to test data rates, etc. this is not the method you should use.

I have also experienced Layer 1 (Physical Layer) issues where the cable was fine but the female port (The jack on your Ethernet card or motherboard) had bent risers that were not making good contact with the Ethernet connector.

Other than continuity (Does point "A" have an electrical connection to point "B") the electrical test you would be able to perform with a multimeter would have nothing to do with data rates.

PoE testing is best observed at the device, switch, or router because part of the IEEE standard for PoE requires an end-point to negotiate with the end supplying power and no current will be present if you just connect a test lead.