You may use a gerund after "arrange." That usage is appropriate when the gerund is describing a generic activity.
Dolly arranges the flowers at the church.
Dolly arranges the cooking for the annual church picnic.
The two sentences have a parallel structure.
Mary arranged for us to stay at her mother's vacation home.
Here we are discussing a specific action. In this case, the infinitive is appropriate.
I would not say that any of your six examples necessarily violates English grammar.
Example 1 is grammatical and idiomatic.
The meanings of examples 2 and 5 are unclear. in some contexts, one or the other may be idiomatic, but the contexts may not be the same for both examples.
Example 3 is not natural (at Least not in American English): "for us to stay" is idiomatic. Example 6 is also not natural because of the placement of "for us."
Example 4 would be idiomatic in certain contexts, but it does not mean what example 1 does.