Question 1. I am supposing that for jQuery, tasks are limited to the following list of APIs. Is that so?
jQuery Core, Selectors, Attributes, Traversing, Manipulation, CSS, Events, Effects, Ajax, Utilities, jQuery UI
Question 2. Is jQuery limited only to DHTML, user interfaces and AJAX? Does that mean that it leaves outside other stuff like audio/image coding/decoding in the browser (with array buffers and data views), or is there a general way to handle every HTML label and CSS property, but in a generic way, without tasks considerably more "advanced" like the ones I mentioned about audio/image?
Question 3. What are the main reasons, if any, to use jQuery other than accelerating the development of user interfaces and not having to worry about the programming cross-browser differences?
Question 4. What are some of the relevant specific elements that differ across browsers, or are there too many to be mentioned, even as the big picture?
Question 5. Should there be a "Hello World" for each API aspect? Or if they do exist, are all of them clearly contained in the jQuery official tutorials and API references? What are there some clear suggestions that are friendly for complete jQuery beginners?
Question 6. What would be the better approach to study and understand jQuery's code, and the references to understand the algorithms and syntax used, to know exactly all it can do and how it is done, and avoid having to guess time and again how to do even the simplest things with jQuery?
In short: no. jQuery comes with a comprehensive set of tools to ease and quicken the development of front end HTML applications. On top of the jQuery core library there is a plug-in system and a large repository of plug-ins that extend the functionality of the core library and provide ui tools not found in the core plugins.jquery.com (currently offline) www.jqueryplugins.com
jQuery is a front end language and relies on the browser to provide and the HTTP stack to provide a lot of the functionality you mention (compressing/decompressing streams).
You are learning / contributing to what is now an industry standard library used in web development, thus maintaining/improving your own skill set as a programmer and making you more valuable to your current/future employer.
These are too detailed and too many to list and range from differences in handling CSS
to differences in JS functionality such as requestAnimationFrame
... the list goes on and on and on. The level of support for H.264
and WebGL
is currently one big issue with browsers. The best way to deal with this is not to try to acquire a complete picture of the differences as browser vendors release updates every few months and the list is constantly changing. Start coding and come back to SO when you encounter problems. In addition to quirksmode.com already mentioned by @Petah have a look at http://www.alistapart.com/ they have excellent resources cross browser compatibility not only with JS but also CSS.
I haven't gone through the complete API docs of jQuery but I have found the docs to provide very good simple and sometimes not so simple tutorial and Hello World examples of how the method calls are used and what they do. These combined with an general understanding of JS and the design methodology of jQuery is enough. For a complete beginner start with How jQuery Works and jQuery Tutorials (scroll down the page for jQuery API tuts) along with the Mozilla DevNet articles on JavaScript as you will need these to understand some JS specific issues.
There is no easy or quick to understand fully all that jQuery - or any library - can and can't do. To get there you need start by using the library, looking through the source code and eventually contribute to all the modules of the project - basically you need to become THE developer for the project. It's debatable if even then you'll know all of it. Which major software project have you been involved in where you alone know all of the entire project and all that it can do. A good place to start is to understand javascript, Mozilla DevNet provides some great resources for that.