Does the family who asks first get it?

Asking first is a perfectly fair way to settle a week, as long as who asked first can be established. On a shared calendar it cannot. The timestamp is buried where nobody can read it, and an entry somebody edited looks the same as one nobody touched, so the arrangement quietly sorts households by who checks the calendar most often.

A request in Mooring is dated when it is made and visible to the household who made it and to whoever answers it. That one fact does most of the work, whatever arrangement the family has settled on top of it.

Can two families take the same week?

More often than the calendar suggests. Plenty of the contested weeks would have suited two households perfectly well in a house that sleeps them both, and it was only an entry that made the week look taken.

Mooring asks on every request whether the stay wants the house to itself, so a week two families would have shared is offered rather than lost. It is on every plan, the free one included, and it is the one question no general-purpose calendar thinks to ask, because no general-purpose calendar knows a house has a capacity.

What if somebody's plans change?

They will, and a peak week given back late is worth much less than one given back early. The thing that slows it down is almost never reluctance. It is that letting go of a week means messaging the person who keeps the diary and waiting to hear whether it landed.

In Mooring a guest can withdraw a request before it is answered and cancel a stay after it, and the week reopens on the calendar everyone is already looking at. Nobody has to be told, because the availability everybody sees has already changed.