It's like "cooks-do", as in the sentence "Cooks do cooking for their living". Actually, it's the literal transcription of some German slang meaning "Look at that!".
Yes. qooxdoo is Open Source, dual-licensed under LGPL/EPL, i.e. the "GNU Lesser General Public License (LGPL)" and the "Eclipse Public License (EPL)". As a recipient of qooxdoo, you may choose which license to receive the code under. More licensing info.
Yes. Always has, always will. It is a true Open Source project: standard OSS licensing, public code repositories, public bug tracking, contributor-friendly.
No. It was registered as an Open Source project at SourceForge on January 2005, one month before the term "AJAX" was even coined. It is constantly being developed and improved, which makes it one of the most mature and innovative web application frameworks today.
Yes. Its solid programming model and tooling allows for creating large-scale applications. To get an idea, see some user-contributed examples (even if that's only the tip of an iceberg).
Use the general mailing list. Many active users as well as the core developers try to answer all your questions. There is no commercial support currently available, but there are several companies listening to the mailing list to offer qooxdoo-related contracting.
While several books contain sections on qooxdoo or references to it, there is only one printed book which is specifically about qooxdoo. Unfortunately it's now outdated because it was written towards qooxdoo v1.4, at a time when only (what we call now) qx.Desktop existed: qooxdoo - Develop Rich Internet Applications (RIA) with qooxdoo. Therefore we recommend the qooxdoo manual instead with its approx. 400 pages. It's maintained by the core team and available for free: as an online HTML version and a PDF to download.
A qooxdoo application runs in all major web browsers. For a typical qx.Desktop application that means: with identical look & feel.
No. If you already have an existing backend that serves HTTP (or HTTPS) requests, it's probably fine to continue using it. Optionally qooxdoo offers several RPC servers for an elegant client-server communication. BTW, during development of your client application the local file system often is sufficient, without the need to use a real server.
No. Well, it depends …
For creating qx.Desktop or qx.Mobile apps it is much more comprehensive than a typical JS library like jQuery. It's also not just a "library", but a "framework": it comes with a large set of high-quality widgets and features a powerful, platform-independent tool chain.
For creating rather DOM-oriented qx.Website solutions, though, it is also possible to leverage qooxdoo as a low-level library.
As with other typical JS libraries, much of jQuery's features are a subset of the qooxdoo framework. To allow for a jQuery-like programming style, an API very similar to jQuery is provided.
There is no official GUI builder that comes with the framework. But there are some community-driven attempts to create such GUI builders.
Please get involved and participate. Check out How to contribute to qooxdoo!