There is no real answer, as anyone could make a new kind of truss rod, for example a licorice truss rod. Perhaps you mean that a truss rod could be under stress or under compression, or be designed for both.

No, they do not require a truss rod. Some of the Banjoleles have them installed, but they aren't required.

No. Some less expensive guitars do not have truss rods in their necks. Any guitar that is more than a toy should have a truss rod, with out the truss rod if your guitar got into some moisture and the neck bowed,you would not be able to straighten the neck out.

You can't!

It's a long rod lodged up in the neck of a guitar used for keeping the neck of the guitar straight.

The truss rod is used to allow the neck to bend either way in response to the tension of the strings. Without it, there would be no steel string guitars, the neck wouldn't be able to handle the tension of the strings. Many beginners think that adjusting the truss rod will lower the "action" or how high the strings are from the fret board of the guitar. This is not entirely true and it is probably more of a Nut and saddle/bridge adjustment that is needed.