I have seen some cases where して is placed before a する verb:
Is this regular practice? Is this grammatically correct? Can this be done with other conjugations of する?
What you're seeing is not お出かけする written in a strange order, but a fixed adverbial phrase みんなして ("get together and (do ～)", "with many others", 一緒になって, 集まって). 大好評 is not a suru-verb in the first place. みんなして doesn't have to be followed by another suru-verb. For example みんなして同じ事を言う means everyone says the same thing almost at the same time (=口を揃えて言う).
You cannot use みんな as a regular suru-verb. みんなして is a fixed expression derived from archaic Japanese, and it always appears exactly in this form. In archaic Japanese, して was used like a standalone particle similar to で in modern Japanese. See this and this. 三人して meant 三人で, 血して書く meant 血で書く/血を使って書く. You may still see this type of して in literary sentences. みんなして doesn't sound particularly archaic to me.
二人して is also used in modern casual conversations (e.g., ねえ、二人して何やってるの？).
して is the verb する which have been conjugated in its て form to connect the 2 parts of the sentences. The fact that the second sentence part of your sentence starts with a する verb or not is pretty much irrelevant.
Do like that.
Do it like that, it's safer.
Other way you can conjugate する in such a way.
If you do it like that, it's safe.