There is an easier way to copy+paste in Cygwin (found it just today, after 3 months of torment).

Open a Cygwin shell -> right click header -> Properties -> Options Tab (upper left corner) -> enable "QuickEdit Mode" -> press Ok ->

check "Save properties for future windows with same title" -> press Ok.

A good idea would be to close all the Cygwin shells and then, in the future, when you'll open another Cygwin shell, you will be able to select text in the Cygwin window. After you have selected the text, press right click. Then go to a Windows text editor (like Notepad or Wordpad), and paste the text there. Similarily, you can copy some text in Windows, then go in the Cygwin shell, right click and you should have the text pasted there.

There are still improvements that could be made, like selecting the text and pressing Del or Backspace to delete it, but hey, if puTTY hasn't got that feature, i really don't need it :D.