1. Make sure your verb endings are consistent. If you end your first sentence with 頼んだ, you should make an effort to use the だ ending for all your sentences.
2. I'd break up the first sentence into two sentences, because I somehow feel you've got a lot of information in that one sentence. 予定だった feels like a natural ending place. 私の友達全員 already sounds as if they are excluding you. Might be more natural to say 友達と一緒に (leave the 私の off because its understood). Or maybe 友達とみんなで
3. And for the next bit, 私は通りがかった友達の一人 sounds a bit too much like English translated one-for-one into Japanese. 私はその一人に or 仲のいい一人に etc., might be more natural.
4. 迎えに来て → 連れって行ってくれるようにお願いした or 乗せてくれるように might be more natural in this first sentence. (Again, I'm not a native speaker so YMMV). For some reason I find 迎えに来る not as good in this case as 連れって行く or 乗せて行く. But for sure it is OK to use it. I just find the other ways somehow more fitting.
5. I feel it would be more natural to say しかし、金曜日の夜、遅くなっても彼は迎えに来てもらえなかった (or 彼は現れなかった = he didn't show up). Then explain why - because it feels like the "punch line" of the story is that he thought you were joking. So it feels like that should be at the end.
彼に連絡して「どこにいるの?」と聞いたら、彼はすでにみんなと一緒にいたんだって。
どうも私がお願いした事は冗談だと思ってたらしい
or something like that.