Q:

Which came first: いや (the interjection), or 嫌がる and similar words?

I assume that the interjection いや is cognate with a bunch of other words beginning with iya- such as 嫌がる、卑しい, perhaps via the semantics of 'disagreeable'.
But did the interjection come first and get lexicalised into verbs and adjectives? Or does the interjection derive from the other forms? Which is the earlier form?

A:

According to the entry of いや in 日本国語大辞典:

否定の感動詞「いな」が中世に「いや」になったと見られる。肯定否定の対応としては、狂言などでは「無いか？」「いや、ござらぬ」のように、否定的な疑問に対して、後続語句に対応した使い方をすることが多かった。

The interjection sense first appears in 1254:

＊古今著聞集〔１２５４〕一六・五二九「『〈略〉件（くだんの）田は相違あるまじ』などいへば、権守とりもあへず『いや田におきては、はやくとられぬ』といひたりけるをかしさこそ」

And the adjective in 1477:

＊史記抄〔１４７７〕八・孝景本紀「先帝の人をわづらはす事をいやにをぼしめしたほどにとてか」

It suggests that the adjective usage of いや (with all its derivation) is a secondary development.
While いや derives from an older form いな in the middle ages, いやしい dates back to 日本書紀, that means those two are not cognate.

＊日本書紀〔７２０〕景行二七年一二月（北野本訓）「是を以て、賤（イヤシキ）賊（あた）の陋（いや）しき口を以て尊号（みな）を奉らむ」

As an aside, いな also generated a derived verb, namely いなぶ > いなむ "deny, negate", which has been widely used till now.

