Let me make sure I understand.
You have a server (dev) that had CF9.0.2 installed, and upgraded to the trial version of CF2016 (I'm never a fan of upgrading over one iteration - there are usually problems.) But you're trying to use the CF2016 Tomcat server to run both CF2016 _and_ CF9.0.2?
I don't think 9.0.2 will run on Tomcat. I don't know that much about it, but I think that 9.0.2 and earlier will only work on JRUN. I don't know the internal differences between JRUN and Tomcat, but if there are any coding differences, then my assumption is (most likely) correct.
If you can completely remove CF2016 (including the Tomcat), and install CF2016 to, say, a completely different drive (including the Tomcat), and keep 9.0.2 aimed at JRUN and 2016 aimed at Tomcat, you might be able to get these to run simultaneously on the same server.
Otherwise, your best bet would be to create another dev server, install CF2016 to it, and just migrate your code over from the old dev server.
VMs are quite handy for this kind of process.
UPDATE: My personal and professional opinion on FLEX and SWF.. don't use them. Flash is on its way out, has been for a while, mostly due to security concerns. You'd be much better off leveraging HTML5 for stuff like that, and if you're concerned about legacy browsers, you can code some kind of graceful way of prompting the user to upgrade to a more modern browser. Just my two cents.
UPDATE2: CFDUMP can be your best friend, when it comes to troubleshooting issues.
I have both CF9.02 and 2016 installed on the same box. CF9.02 is on IIS port 443 (HTTPS/SSL) and 2016 is on Tomcat using port 8500 (HTTP no encryption).
The legacy apps are up for a re-write, but for the short term we need to see if we can get the LFEX apps to run on CF 2016.
Since your legacy CF is going through SSL, and CF2016 isn't, my first thought is that this might be an encryption issue.
But to be sure what is happening, use CFTRY, CFCATCH, and CFDUMP to determine precisely what is causing this issue.
I did a quick Google search for CF2016 and FLEX, and the only relevant link that I found was this one, which I hope will help. Outside of that, I cannot think of anything else.