I've shyed away from using tmux and Vim. For some reason, when I use tmux and Vim, if I 'copy' a snippet of text (using my mouse/cursor higlight -> right-click -> copy) it treats all of the empty space a actual "space" characters when I 'paste'. Why does it do this?
You can let vim catch the mouse clicks / drags and have it manage you X11 clipboard (start looking at vim help for :set mouse=...). Personally I prefer to not have vim mess with my mouse and X11.
I think it's better to just use vim's horizontal selection ("visual line", Shift+v) and use yank (y key) or filter it into an extern command like "xlip -i -selection clipboard" to copy the highlighted text into the clipboard.