We have a 2-server (web and app server plus a database server) SharePoint 2013 farm. We would like to add SharePoint-integrated mode SSRS 2012 to it. We were thinking of installing SSRS SharePoint mode on a different SQL box (existing database server). However, It seems like in SQL Server 2012, Reporting Services SharePoint mode needs to be installed on the SharePoint app server and cannot be installed on any database server. Is that right? If that is the case, does that mean we need to license the SharePoint server for SQL Server as well?
If you want to use SSRS in SharePoint mode, then you have to install it on the SharePoint server. The SQL server can be on a different machine.
Licensing questions of this type are off-topic here. You should ask your Microsoft reseller. That's what you pay them (a percentage) for.