On which node first transaction will be executed in Always-on setup?
1) From Overview of AlwaysOn Availability Groups (SQL Server)


  The primary replica sends transaction log records of each primary database to every secondary database. Every secondary replica caches the transaction log records (hardens the log) and then applies them to its corresponding secondary database.


So, primary first, then secondary.

2) Are you asking how node 1 is brought back on line? That's what your operations people and DBAs are paid their huge salaries to do.

Alerts would be a good idea. Maybe even a very good idea.

If node 1 dies AlwaysOn will re-direct connects to node 2, transparently to the application. In-flight work may receive an error. New connections will be routed to node 2. This is the HA and DR in action. When node 1 comes back you can continue running on node 2 as primary and have node 1 as secondary, or organise a managed failback at a time of your choice. If node 2 dies while node 1 is still down, then you have a problem.

3) Active-passive has the secondary as "warm": it is ready to become the primary when required but cannot do any application work until then. Active-active has a "hot" secondary: it can process read-only transaction from the application, and be used for backups and such like. IIRC one license is needed for active-passive and two for active-active.