In a SQL Server AlwaysOn AG synchronous mode scenario, if the secondary node goes down, is the primary down as well (since it cannot write to and commit to the secondary)?
No. If the secondary drops, the primary node will continue operation. However, you need to be aware of how much disk space the queued transactions will consume. If you lose the second node for an extended period of time, it may make sense to evict it from the cluster so that the primary node won't queue the transactions.