is it actually possible to do damage to the analog input sockets of Orville if, instead of using unbalanced jack plugs, I used balanced ones?

To set your mind at rest – highly unlikely.

There is actually no such distinction as a balanced or unbalanced jack plug – these refer to the way the plug is connected – balanced has to be TRS, unbalanced can be either TS or TRS.

If you use the wrong one, in may hum and buzz, or have no signal, but is unlikely to do any damage (I say unlikely, because you could potentially have a major AC power ground problem, or lightning strikes, but these are issues way beyond your jack plugs).