I trying to put my Mac's data in order. I have many rc-files at my root such as .vimrc, .srceenrc and .bashrc.
I would like to put these files to the specific folders such as .vimrc and .screenrc to ~/bin/coding and .bashrc then again to ~/bin/shells.
How can you determine where these rc-files must be?
Seriously, you should leave them where they are. Applications will be looking for them in specific locations (probably your $HOME directory which is not root, by the way, or shouldn't be). This is a very old UNIX convention that you should attempt to change only if you fully understand the consequences.
Not meaning to sound condescending but your error in naming your home directory as your root directory seems to indicate your knowledge level of how it all works is less than it should be to understand those consequences (apologies if that offends you, I agonized over the best way to say it - what I mean is that you should tread carefully).
If you move them, you will have to ensure you run the applications that use them with their paths fully specified, and some applications may not let you do that.
They all start with "." so that they're hidden to the normal ls commands and, if you're using a graphical file browser, there should be a way to hide them there as well (such as the Gnome File Manager CTRL-H).
Configuration of a program is both defined at system-level and user-level, you can tweak the user-level one, which resides in your home, to help you in what you need.
No need to group them in subfolders as you said: leaving them in your home (not root) is following the convention everybody uses, rc-files usually stay there after the program has been uninstalled, so if some day you make a fresh install you'll find the application configured as you left it.
Also, by leaving them in your home, you can bring your own home folder to another system and have the environment set as you like it.