Saying you can replace と with てお is strange. てお is not a meaningful unit of anything in Japanese. ておく is contracted to とく, ておいて to といて.
It's like in English you can say "cannot" can be contracted to "can't". You wouldn't say "nnot" can be contracted to "n't" because neither "nnot" nor "n't" are meaningful units of anything in the English language.
オーライをあげる is not really natural, no. At least in Japanese, you don't say "結果オーライをあげる" ("give someone an 'everything is fine'). You say "オーライにしてあげる", i.e. "make it so that all's well" (do someone the favor of seeing the situation as "all's well that ends well"). You can't completely capture why the former is unnatural and the latter is natural through English translation alone. It's just about what's a natural expression in Japanese and what isn't.
Again, you're showing a tendency to make judgments based solely on English equivalents, and that's not a smart idea. Japanese is its own language, and what is and isn't natural or idiomatic in Japanese is not solely based on whether or not the English equivalent is natural or not.