Do I need SQL Server 2016 Enterprise Edition for installing it on a Microsoft failover cluster?
No. You can setup a two node failover cluster with the standard edition of SQL Server. Any more than that and you need the enterprise edition.
No, you can run a two node Always On failover cluster instance using Standard Edition. See the "RDBMS High Availability" section of https://msdn.microsoft.com/en-us/library/cc645993.aspx