Edit: After writing this answer, I checked a new Linux Mint installation. For you, there should be a simple solution. I'll also leave some of the original information in case those instructions might be useful to others in a similar situation.
For a Linux Mint user
I just checked a system with a new Mint 17.3 installation. Mint comes bundled with Firefox. The Mint repo has the latest version of Firefox. It was installed with the latest version of the Adobe Flash player, 11.2.202.554, so that is what is available in the Mint repo. There is no reason for you to have to mess with a tarball from Adobe. You should be able to simply update the Flash player from the package manager, and that should have been offered as an automated update.
Fix your system
Use your package manager to uninstall Firefox.
Then check your package manager for the Flash player plug-in in case removing Firefox didn't remove a version you added. If you find one, use the package manager to delete it.
Then go back over what you did manually and delete any files you copied in if they have not already been deleted.
Use your package manager to install Firefox.
This should give you a clean install, including the latest version of the Flash player. You should be done.
For users of Linux other than Mint
To the best of my recollection, Firefox is bundled with the Adobe Flash player in every distro I've seen that offers Firefox. It may not be the latest version of Firefox or the Flash plug-in, but it will be a working installation that you can update. So an easy first step would be to follow the instructions for Mint, above, to see if that fixes your problem or gets you to an installation you can update. If that doesn't do it, or you need to install a newer version of the Flash player than your repo provides, see below (from original answer).
I can see a couple of potential problems in the procedure you describe.
Location
There's some variability in where different distros put things. You mention two directories where you placed libflashplayer.so. On my own system (Debian), I didn't find a .mozilla/plugins folder, but found four other locations so far that appeared to contain the actual Flash player:
/usr/lib/flashplugin-nonfree/, which contained a year-old version (which would explain why Firefox kept reporting a blocked old version even though I confirmed that a newer one was installed)./usr/lib/mozilla/plugins/contains just a symlink, which points to the file in:/etc/alternatives/, which, in turn, contains just a symlink that points to the file in:/opt/mint-flashplugin-11/.
If Firefox isn't finding it, you may be missing the place where it looks on your system. If Firefox recognizes the existence of the plug-in, enter this URL: about:plugins. Scroll to Shockwave Flash and it will show the installation directories.
On mine, it listed /usr/lib/flashplugin-nonfree/ (my answering this question solved a problem on my own system), and /opt/mint-flashplugin-11/. On the new Mint 17.3 installation, only one directory is listed: /usr/lib/adobe-flashplugin/ (which is also the case for Mint 17.1; I couldn't confirm it back to 17, but it's a good bet it's the same on your system).
Symlinks
By manually copying the plug-in into /usr/lib/mozilla/plugins/, you may have broken a symlink chain and replaced it with mixed versions of the plug-in. I'll have to think about whether that could explain why the plug-in wasn't found, even after you installed a copy of the Flash player with your package manager. That chain will be unique to your distro, so there isn't a simple way to fix it. The most direct way would be to find a copy of the original symlink file. I would expect that following the procedure described for Mint users would replace any required symlinks.
Meta Data
Installing from the tarball, there is also a collection of meta files that go in various /usr directories. This question didn't specifically mention installing those.
Adobe problems
As Xen2050 mentions. Adobe is dropping support for Linux; Version 11.2 is the last one. I had been wrestling with Flash for months. The version available through my package manager (11.2.202.540) still showed up as out-of-date (and as mentioned above, was reported incorrectly). The APT package for Ubuntu-based systems available from the Adobe download page has had a bad URL for more than half a year. Adobe doesn't seem too invested in fixing problems for Linux.
However, your question provided the impetus to test the generic Linux .tar.cz version from Adobe, which contains the most current release (11.2.202.554). I installed that manually, per the included readme file and the locations designated in Firefox's about:plugins page. That worked.
Fixing your system
If all of this was done in order to manually install a newer version of the Flash player, this may help. On your system, it looks like you installed the version available from your repo on top of the possibly newer version you installed first from the tarball, and perhaps in the wrong location. As a simple first step, check Firefox for the correct locations and just install the latest version using the tarball, following the readme instructions.
An alternative to the about:plugins page for identifying the directories would be to uninstall the Flash player plus Firefox, itself, using your package manager. Then reinstall Firefox. Firefox typically comes with the Flash player, and this would get you a properly installed setup, which you could then manually update.
When installing from the tarball, you may need to use sudo to copy the files into the system directories, which normally require root permissions. If you didn't use sudo, part of your problem could be that it didn't actually copy the file.
You will need to restart Firefox for it to load the new player.
If that doesn't work, You may need to clean up what's there. The Flash player does not have a component in the profile directory (/home/.mozilla/...), so if you messed with that, you may need to restore the profile (see Xen2050's answer).
If the multiple actions you performed corrupted the installation, follow the instructions in codemonk113's answer.
If you reinstall from the repo, you may not have the most current version, and Firefox may complain because Mozilla has been pretty aggressive against vulnerable Flash players. If you want the most current version, you may need to install it from the tarball.