Since version 3.2.0 BEdita is fully open source and released under GNU Lesser General Publica License.
For previous versions (3.1.x and 3.0.x) BEdita is available in two different versions: a free/open source AGPL v3 version for creating free/open source applications and a commercial/proprietary version to build commercial/proprietary solutions. If you intend to use the free/open source version beware to respect the license terms, in general take a look a the GNU FAQ and this ovverview on OSS Watch.
Of course you are free to build proprietary and free software solutions for your client. Look at Licensing for more information and useful links.
BEdita has been designed/created as a framework, not a real CMS: it handles many types of contents in an advanced way, but there's no builtin default frontend.
Frontend applications could vary, in our opinion there is not a predictable way to build one suitable for everybody's needs: it may not to be a Web application at all, for instance. Anyway you can find some useful frontend examples in the frontends directory and on github
BEdita has a web/wizardinstaller since 3.2 release. We strongly suggest you to read carefully the INSTALL file, the online documentation and specially this article if you have any issues. Since BEdita is built upon the CakePHP framework your setup problems could be Cake-related, so please take a look at the documentation and at the various cake resources. If you are deploying BEdita on a server in housing/hosting read carefully the minimum requirements (INSTALL file and requirements), not always they are fullfilled. If you can't solve your problem write a message in the dedicated section of our forum.