Is the FLV video player from Firefox?

Sounds suspicious. I think it might be time to cast a skeptical eye on your add-ons.

Try disabling ALL nonessential or unrecognized add-ons on these two tabs:

orange Firefox button (or Tools menu) > Add-ons > Plugins category
orange Firefox button (or Tools menu) > Add-ons > Extensions category

By nonessential I mean you could live without it for 48 hours.

Usually a link will appear above at least one disabled extension to restart Firefox. You can complete your work on the tab and click one of the links as the last step.

Any improvement?

You can also check the target line in the Firefox desktop shortcut.

Did you try to clear the cache and check for unfinished downloads in the Downloads Manager that might get (re)started?

Please just tell me if FLV is good (or at least is it part of Firefox)?

no, flv player is not a part of mozilla firefox & you shouldn't download it, just because a pop-up says so.

your computer seems to be infested with adware. please try to reset firefox, and also go to the windows control panel / programs and remove all toolbars or potentially unwanted software from there and run a full scan of your system with the security software that you have in place and different other tools like the free version of malwarebytes & adwcleaner.

Remove a toolbar that has taken over your Firefox search or home page Troubleshoot Firefox issues caused by malware

I don't think there's any way to answer your question about whether a particular download is trustworthy without actually knowing exactly which of the many players it might be. Assuming it even is an FLV player and not something else...