Does visual studio (2015 or above) needs to be installed in a server in order to transfer/create SSIS packages there? At work we have an old server and we're moving everything from there to our new server. We're a very small team, and one of my team-members says without visual studio 2015 in the new server he can not transfer/create SSIS packages there. Any thoughts on this?
Visual Studio does not need to be installed on the SQL Server.  SSIS packages built with SQL Server Data Tools can be exported to .ispac files, which can then be deployed to servers using SQL Server Management Studio.  

See Deploy Integration Services (SSIS) Projects and Packages for more details.

One thing to note is that I've found that deploying .ispac files to SQL Server 2016 servers requires SSMS 2016 for some packages.  Packages deployed with later versions of SSMS end up with versioning errors (Source - ODBC is not compatible with this version of the DataFlow).