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I’m with you in believing that this is going to be a typical よつば mishmash!
I have no idea how it fits together, but the “Yotsubato! Volume One Vocabulary List”, which you can find a link for at the top of this thread, does a great job of breaking it down like this:
どーぞ is an informal form of どうぞ and, like you say, used to say “please, by all means”
あがりたまえ according to this list, is an informal form of 上がる/あがる, meaning “come on in” (informal!)
And then the list says おまかい is not a word, which is why Fuuka then repeats it, though the list suggests she may have meant こまかい (small), which strikes me as correct as many people will say apologetically “come in, come on in, though I’m afraid it’s very small”.
Like I say, all this comes from the Vocab List which you can find a link to at the top of this thread.