Can Visual Studio 2013 be installed on a server which hosts SharePoint 2010? Are there any potential issues of installing it?
As a short answer: Yes you can install and create a
SharePoint 2010 Solution using
Visual Studio 2013 without any issue.
Note: Don't forget to install
Office/SharePoint Template tools to show
Office/SharePoint Template for more details check Couldn’t find Office/SharePoint template in Visual Studio Community 2013
Visual Studio 2013 can be used to create SharePoint 2013 solutions. You can use VS 2012 for SharePoint 2010 solution. But you can open the solutions using VS 2013.
You ca find more info about VS 2013 here.
But there are few posts says it is possible with VS 2013 premium edition this we haven`t tried.