Your host has the right to decide if you can leave your dog alone in the rental. Some hosts will explicitly state in the house rules that you cannot leave your dog alone.
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If possible, dogs older than 6 months should not be left alone for longer than 4 hours at a time. If this is difficult, the absolute maximum time for them to be alone is 8 hours, but this is only recommended if your dog has a way to get outside for a bathroom break.
Airbnb's policy on guests leaves it up to the host to specify and encourages hosts to be clear about their limits. Hosts can state the number of people, cost per person, and the number of beds in the property. However, there is a limit of 16 people, to discourage large parties.