Can I move all dotfiles from ~ to ~/.config?

Short answer: no.

You can't move your dotfiles to ~/.config and expect programs to somehow infer that no dotfiles under ~ means they could be elsewhere.

Therefore, you ought to explicitly tell every program to look at ~/.config. (I guess in most cases this is done by setting some environment variable.)

Even then, many programs don't even envision such a possibility; so you'd still have some dotfile under ~.

No you can't move ~/.* to ~/.config

Most files and directories beginning with . in the top level home directory (/home/$USER) are controlled by applications doing things on your behalf.

Folders within such as .../Documents, .../Pictures, .../Downloads, etc. are almost always your exclusive domain. The best practice is to never put your own files into /home/$USER(~). Then you aren't inconvenienced by searching through all the . files and directories that you don't control.

Using the good ship Nautilus to navigate troubled waters

I'm a self-confessed CLF (Command Line Freak) dating back to IBM PC-DOS days. After a few years though I've come to admire the efficiency and power of Nautilus.

Within the Nautilus file manager you can hide these files and directories from your eyes. Go to the top bar Files menu and select Edit then Preferences and unset the option on this screen:

In answering this question I did the above myself and instantly the main Nautilus window repainted. To my surprise I discovered a dozen of my own files that had been hidden within all the system files. I moved them to appropriate sub-directories or deleted them.

Nautilus hot key / keyboard shortcut

Press Ctrl+H to toggle hidden files / directories viewing on and off. This doesn't change the permanent settings described in the previous section. It only lasts for your current session but saves you the inconvenience of changing the configuration for one time views.

Many thanks to commentators below (@videonauth and @utf-8) for providing information on this section.