Oh how fun. Don't you love troubleshooting?
I think I I were in your shoes, I'd begin by simply closing the project you are in and creating a new empty project. Add a few topics. See if the lag still exists after playing with it for a while. If so, you know it's got something to do with RoboHelp and your setup. And if there seems to be no lag, my own suspicion is something to do with the fact source control is in the mix. At that point, further troubleshooting would be in order.
... or try one of the sample projects that ship with RoboHelp.
There is no lag when creating or working on a new blank project. The Adobe support person tried that when they worked on my computer. Sorry, should've mentioned that.
We do not have the option of not using source control (SVN) with our help so I hope that's not the issue. If that's the case why would RH11 work great with SVN and RH 2015 does not?
What other troubleshooting can I try?
I'm not a specialist for your question, but I might be able to provide some more reference information. I'm using RH 2015 with TortoiseSVN on Windows 7, 8 GB RAM, also with several large projects (size +1000 topics). I don't have any lag issues, neither when opening a project nor when typing or saving. So I don't think it's a general problem of using version control and RH 2015. More likely your problem is caused by a mix of several issues.
Any other ideas on how to troubleshoot this situation?
I assume that you didn't also change operating system, etc when you upgraded to RH2015? If you did, perhaps you need to check you have 32-/64-bit versions of (sorry, mental blank) the api/dll/thingy that RH uses to talk to SVN. ( Just a thought experiment, so may be completely irrelevant. The thing I'm thinking of is specifically for TFS - the MSSCCI Provider, but possibly there is a similar thing for SVN).
Or anther thing to check is your antivirus. Many years ago I had problems with very slow check-in/out and it turned out to be the antivirus scanning every file multiple times. The solution was to make sure the AV only scanned once, rather than on both actions - I think it was only scan on check-out, but possibly vice versa.