Are the Cygwin tools free software?

Yes. Parts are GNU software (gcc, gas, ld, etc.), parts are covered by the standard X11 license, some of it is public domain, some of it was written by Red Hat and placed under the GNU General Public License (GPL) or GNU Lesser General Public License (LGPL). None of it is shareware. You don't have to pay anyone to use it but you should be sure to read the copyright section of the FAQ for more information on how the GNU GPL may affect your use of these tools.