배어들다

배어들다 (cling) is 배어들어요 in the polite 해요체, 배어듭니다 in formal 합니다체, 배어들어 in casual 반말 and 배어든다 in the plain written style. The past is 배어들었어요.

배어들었어요 in everyday polite speech, 배어들었습니다 formally. It is built from the 아/어 form 배어들어 plus ㅆ, which is why an irregular stem shows its irregularity here.

Yes — it is ㄹ dropping (ㄹ 탈락). ㄹ drops before ㄴ, ㅂ and ㅅ: 살다 → 삽니다. Regular, but universal. That is why the stem looks different in 배어들어 and 배어드니 but unchanged in 배어듭니다.

It carries no TOPIK grade. That is a fact about the published word lists rather than about the word — it conjugates by exactly the same rules as everything else.