So far, every program was either installable by .deb file or through Ubuntu Software Something. What it did is create a nice file, which I could search up, which had a nice looking icon, and I could drag this icon, onto bar and it would be saved as favorite. I forgot to mention I use Gnome. But then, I wanted to download TeamSpeak 3, Skype-ish software that I find better when it comes to gaming a team coordination. It delivered me a .run file. Now, I've Googled a lot and I managed to find a way to get it running (no wonder, I saw post saying that .run is like .exe but from Linux), but, I would also like to have TeamSpeak 3 on my favorite bar, not just a clickable exeuctable without icon, on the desktop. I want full favorite program, with it's icon in my favorite bar, is there a way to do it, or I have to get along that .run files work this way?

@edit - Even programs that were "cheated" on, and that were forced to work on Linux by all kinds of tricks while they're Windows installers, managed to drop an icon that I could've put onto bar and were runable by just one click.

You should be able to right-click on the panel, and select the "add to panel" option to do what you want.

Gnome doesn't consider "free" space a "free" space. Whenever I right click, it right-click selects application and it asks me if I want it remove it or start new window with it. When going to going to special button "Show Application" it doesn't show anything anymore.

Are you running the same .run-file over again (I confess, I have never used teamspeak)?
A .run-file should put the programme somewhere in your bin- or opt- directories (install). If your installer hasn't done so, you can then create a .desktop-file pointing to the executable.
You can then place this file into /usr/share/applications/ and run "update-desktop-database" your icon will show up in the dash. You can then simply add it to ypur favorites.