I am wondering, if we want to express that sth is left in a condition, for example the condition of being idle, can we write "sth is left idle"? Here, is the verb "left" considered a linking verb?
You can certainly write (and say) "something is left idle". In fact that's exactly the correct way to put it.
"To leave", itself, is not a linking verb, but for the purposes of your sentence, with its passive construction, you can consider "is left" (or "to be left"), together, as a linking verb, with "idle" as the subject complement.
From another perspective, arguably you can also consider "is" (or "to be") as the (linking) verb, and "left idle" as an adjective phrase, and as the subject complement. There are often many ways to parse a sentence.
In this case, however, you might want to consider it from a third angle.
Because "to leave" is transitive in its active form (it takes a direct object, not a subject complement) in the sense under consideration, on its own you shouldn't consider it a linking verb.
But either way you can think of "idle" as resultative; i.e. as a result of being left, the thing remains idle. And if you make your construction active, you can see that "idle" is still resultative, but "to leave" is definitely not a linking verb: "leave something idle". So, for example, she deliberately left the machine idle, where "deliberately" is an adverb of manner, and "idle" is a resultative complement (also called a predicative adjective as object complement).
It's common for an adjective to act as a resultative complement. You can read more about resultatives here: https://en.wikipedia.org/wiki/Resultative.