After this dialogue is displayed, are you able to run Firefox?

If so, please try resetting Firefox. ^{(More Details)}

Not able to run Firefox, it just closes. Therefor resetting not possible that way. I'm writing using Safari..

Sorry to hear that you are not able to run Firefox.
Can you please try starting your operating system in safe mode. From safe mode you can then attempt to open Firefox.

Starting In Safe Mode

1. Ensure that your computer is shut down completely
2. Press the power button
3. Press and hold the Shift key after the startup tones
4. Release the Shift key when the grey Apple logo is displayed

When in safe mode, can you then open Firefox?

If you can not open Firefox when your operating system is in safe mode, it may be because of a damaged Firefox file.
If this is the case, you will need to completely remove Firefox. ^{(Follow These Steps)}

Does this fix your issue? Please let me know.

I've restarted osx in safe mode and firefox was opened in safe mode and it appears to be working. What might I do next to find or solve the problem?

Your problem may be caused by one of your settings, one of your add-ons, or some add-ons fighting each other. Can you list your add-ons and plug-ins?

I restarted osx in normal mode and opened firefox, which has opened normally, it would appear :)........ (doesnt appear to be in safe mode).

This seems a bit curious given that it didn't work at all prior to safe mode start.

Has Firefox reset itself or something?

This is quite curious.
Is Firefox now fully functional when both Firefox and your operating system are in normal mode?