Can we download SSRS 2017 to use with SQL Server 2016? We are using our organization's SQL Server 2014 SP3 servers.  (They also have 2016 SQL Servers, but we elected to skip 2016 and just move to the next SQL servers when they get installed.)
We have a need for the REST API access in SSRS 2017, so we would like to upgrade our current SSRS server to SSRS 2017.  

The Windows server admin just set up a new/test SSRS server for us, but he used SSRS 2016, because that is the latest media that the organization has, and they are of the impression that we can only install SSRS for the SQL server version that we're using.

Am I correct in thinking that it should be possible for us to download (for free) and install SSRS 2017 and then point it to the existing 2016 SQL servers?  Or do we need to purchase a license for SQL Server 2017 to be able to use SSRS 2017?

(This seems similar to existing question SSRS 2016 on SQL Server 2014, but that question and answer seem to focus on SQL Server Express, which is not our situation.)
You can download and install SSRS 2017, and use an existing SQL Server to store the Report Server database.  The databases can be on any supported version of SQL Server, as documented here.

The server running SSRS 2017 must, of course, be licensed per the licensing guide here.

Alternatively you can get this functionality (and more), with Power BI Report Server, which can be acquired via Power BI Premium, or run using SQL Server Enterprise Edition Core Licenses covered by active Software Assurance.