Upgrading SQL Server 2005 to 2016
Since you are migrating to a new server, you don't need to bring SQL Server 2008 into the mix.  

As stated by SQL Server expert Paul Randal, You can upgrade from any version 2005+ to any other version.  Paul states:


  There’s a persistent myth that you cannot upgrade a database to a
  version more that is more then two versions newer.
  
  It’s not true.
  
  You can backup+restore (safest way) or detach+attach (not a safe way)
  from SQL Server 2005 (or higher) to any other newer version. You
  just can’t do an in-place upgrade to the latest version from SQL
  Server 2005.


As for changing the compatibility level, I'd recommend not changing that for several weeks and let the dust settle from the upgrade.  After you're confident that everything is functioning properly from the upgrade, why not bump the compatibility level up to the maximum and monitor performance.  You might even see an improvement in performance.  However, compatibility levels from 120 (SQL Server 2014) and above bring the new cardinality estimator into play, which might present some performance problems.  If you want, you can be more selective in your testing of the new cardinality estimator by using OPTION (QUERYTRACEON 2312) on selected queries.  You can always reduce the level should problems arise.  Do some internet searches on sql server 2014 cardinality estimator issues to familiarize yourself with possible issues.

At some point in the future when you upgrade to a later release of SQL Server, you WILL have to eventually increase the compatibility level.

Finally, check out Upgrade SQL Server for a checklist of items to consider for upgrading SQL Server.

Referencing ALTER DATABASE (Transact-SQL) Compatibility Level:


  From an application perspective, the goal should still be to upgrade
  to the latest compatibility level at some point in time, in order to
  inherit some of the new features, as well as performance improvements
  done in the query optimizer space, but to do so in a controlled way.
  Use the lower compatibility level as a safer migration aid to work
  around version differences, in the behaviors that are controlled by
  the relevant compatibility level setting. For more details, including
  the recommended workflow for upgrading database compatibility level,
  see the Best Practices for upgrading Database Compatibility Level.