Is the last word an adjective or the past participle of a verb in "The student looked at him, perplexed."?
Solution 1:
In your example, only AdjPs work as substitute.
The student looked at him, happy / enraged / confused.
We know they're adjectives because they can be used predicatively where a past participle would not be allowed with verbs like seem, look, sound etc.
The student seemed happy / enraged / confused.
However, we can't substitute a clear past participial clause like the one below.
*The student looked at him, given a fright.
*The student seemed given a fright.
(something) gave the student a fright.
'perplexed' in your example is an adjective phrase functioning as a predicative adjunct according to CaGEL terminology.
It could be expanded:
The student looked at him, perplexed by the boy's insolence.
Which would still work with seem, look, sound, etc.
He sounded perplexed by the boy's insolance.
The meaning here is pretty similar to the clause:
The boy's insolence perplexed him.
But since it designates a state that he's in, and behaves otherwise like an AdjP, it's probably best not to analyze it as a clause.
The only way a preterite form (past tense) would be allowed would be in a coordination.
The student looked at him, went home.
The student looked at him, and went home.
Solution 2:
The student looked at him, perplexed.
In this, perplexed, is a depictive adjective. We can rewrite this as
The student looked at him [and he was/appeared to be] perplexed.
You can compare this to
She arrived at the meeting drunk = She arrived at the meeting [and she was/appeared to be] drunk.
This differs from
She arrived at the meeting drunkly (adverb) = She arrived at the meeting behaving in a manner that indicated she might be drunk.
Your grammar checker accepts "hated" as it is possible, with the appropriate context, for The student looked at him, hated to be a coherent sentence.
There is a pdf document "Resultative and Depictive Constructions in English" by Chang-Su Lee that gives an overview of these constructions.