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In this sentence, it means 'and' or 'because'. You can use the て form to join two sentences or to express a reason, and with nouns/na-adjectives you use で.

I was sick and was absent from school./I was away from school because I was sick.

どうも有り難うございました！

Is this the same rule for using で at the end of な形容詞 and nouns, and くて at the end of い形容詞？

Basically. As Aodh mentioned, it's also like the て form of verbs in compound sentences, e.g.,

病気になって学校をやめた。

You can think of で here as the て form of the copula だ/です.
(In the case of compound sentences--there is never a でいる for present progressive
like the normal ている for usual verbs, though there is a である in written form.)

I think of this usage of で/て as primarily meaning "and," with the possiblilty of giving a reason (i.e., meaning "because").
E.g.,

学校へ行って友達に会った。I went to school and met a friend (not because)

Though in the 病気 example, you're saying "... got sick and quit/took off from school." Here the first part provides the reason, so one could also translate it as "because."