The て form is used at the end but not for a request. It doesn't seem to be one of those cases where the order of the words is simply "inverted" either, so what's the nuance behind the て form in this case?
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Te-form at the end of a sentence can be:
で is usually used to connect to phrases, but when the speaker is too excited about the first part already, s/he wants to affirm it with ね
For that reason, you will keep encountering "sentences" ending with conjunctions and verbs/adjectives in the te-forms for as long as you study Japanese. We call those 「言いさし表現」.
In this case, it's 3. This te-form is used to keep the listener's attention by indicating this is not the end of his story and he has something more to say about it.