running SQL 2012 SSIS packages in recent versions?

SQL Server 2016  - is that a same install like 2012? 2016 is not web based like office 365, correct?

Correct. Its like 2012 only and not Web based.

I created SSIS packages in SQL Server 2012 data tools. Will those SSIS packages run in SQL server 2014 and 2016 data tools?

According to MSDN its possible (https://msdn.microsoft.com/en-us/library/bb522577.aspx)

On an instance of SQL Server 2014, you can import packages from an instance of SQL Server 2008 or from an instance of SQL Server 2012, but you cannot export packages to an instance of SQL Server 2008 or to an instance of SQL Server 2012.

Run packages. You can run SQL Server 2008 Integration Services and SQL Server 2012 Integration Services packages by using the SQL Server 2014 version of the dtexec utility or of SQL Server Agent. Whenever a SQL Server 2014 Integration Services tool loads a package that was developed in SQL Server 2008 or SQL Server 2012, the tool temporarily converts, in memory, the package to the package format that SQL Server 2014 Integration Services (SSIS) uses. If the SQL Server 2008 or SQL Server 2012 package has issues that prevent a successful conversion, the SQL Server 2014 Integration Services tool cannot run the package until those issues are resolved. For more information, see Upgrade Integration Services Packages.