If I have an open wire and a multimeter -- is there a way to check whether there is current and what voltage it is if the wire is not connected to anything, i.e. doesn't have another wire or an appliance to send the current to? Is there a common method whereby I can send the current to somewhere and then put the other probe of the multimeter there to simulate a circuit?
A non-contact voltage tester can detect line-level voltage without needing to complete a circuit. It does this by detecting the alternating electric field that radiates off any “live” household wire. It won’t detect direct current (uncommon in household settings) and will sometimes give false positive readings if the wire being tested isn’t grounded and is close to a live wire.
You can always use your multimeter in the AC Volts mode to test between an unknown wire and a ground. In the modern US, that means put one probe in the central U-shaped hole in the nearest outlet, and touch the other probe to the wire under test. This may also give false readings for the same reason as mentioned above.
By the way, the reason it’s valid to test voltage relative to ground is because that’s the test that matters to personal safety - if there is voltage relative to ground on a wire you touch, you could potentially complete the circuit with painful results!