Is it possible to have traffic routed through a VPN in Firefox but only regular traffic, not stuff like bank websites, Facebook, emails, etc. anywhere where authentication and IP addresses matter for security?
Not easily. A VPN tunnel usually works on a lower level than the browser.
When you connect to a VPN network using a virtual network interface, the OS usually routes all your traffic through this interface by default. The browser has usually no say in this, because it only utilizes whatever the OS provides as the default network route. However, it has to said, that when using WebRTC there might be an IP leak due to the browser poking other interfaces, including the original non-tunneled one.
The way how you could go about setting this up would be a per-site proxy rule using the FoxyProxy extension that would send data through a local proxy depending on the URL mask. This local proxy would be bound to your local VPN tunnel, but by default the VPN tunnel wouldn’t be used for the rest of the applications running on the system.
So, when you connect to a website and it matches the URL mask, the browser would send your request to an ordinary local HTTP proxy and this proxy would then send your data through a VPN tunnel. You might have to setup a local VPN for this and chain the local VPN with your intended VPN tunnel.
EDIT:
Basically, this is the setup you are looking for:
Some VPNs also provide a SOCKs proxy. I believe there are a few add-ons that have URL filters for proxy configurations that you can use.
Well, you could install many of those vpn/proxy addons and disable them for private windows, then use private window to open your bank sites etc