If I can/do install the Dashlane extension, do I remain anonymous?
Dashline is a closed-source, third-party add-on. As the source code isn't available, you have no way of knowing what it's doing. It could be bypassing any proxy settings, including Tor entirely, revealing your real IP address. Add-ons also add to your unique browser fingerprint.
Add-ons and extensions are generally recommended against for exactly these reasons.
The same argument can be used to explain why using vanilla Firefox proxied through Tor - instead of the Tor Browser Bundle - is a bad thing. You don't know how Firefox might by bypassing the proxy. (For the many anonymity-related changes that have gone into the Browser Bundle, see the Browser Bundle Design document. See here for an associated discussion.)
If you really have to install the add-on, then try first allowing add-ons in the
config:addons settings, as per these instructions.