Emulate an older version of chrome within the dev tools
If you just need to check what server returns for old browsers (not full emulation), you can set User-Agent in Network Conditions (for now, chrome 67).
It's not possible to emulate older versions of Chrome from within DevTools, as is possible in IE. It's not very reliable to emulate them anyway. I've used IE 8 emulator in the past, and on occasion it has provided different results from the version I installed on an XP VM.
In theory, it should be possible to install multiple Chrome instances side-by-side because the installation directory is OS user based. However, I don't think the installations are fully isolated from one another. At least in Windows, the application would likely write to the same registry keys. In OSX, you might be alright, provided you modify the user directory path.
An alternative solution would be trying the portable versions of Chrome that appear over the Internet, usually illegally. Everything is self contained in a directory, but it will likely perform differently from normal installations. Check out PortableApps.
I think the best solution would be to set up some VMs for the different environments and browsers you want to test with. You can manually set up these, using VirtualBox or similar. Many hosted providers will allow you to spin up fresh VMs with automated tasks to install applications and services. You have to pay for these though.