1, 2 & 3: We tend to think that we post in a blog first and a category second, but b2evolution actually stores stuff with a category that in turn is attached to a blog. That's sort of important to understand the answers to your questions.

To create similar categories across different blogs you have to actually create the category in each blog. Therefore you might have 5 categories called "dogs", but each one is actually has a different category ID number. There is no way to create one category that will be available in several different blogs. Anything you post (published) will be visible in blog #1, and typically displayed in "most recent on top" order. It won't matter what category or blog - it's simply chronological.

When you make categories and sub-categories in different blogs they are not automagically available for each post you make in any blog. The 'default' way is to show you cats and sub-cats for the blog you're posting in. You can edit your conf/_admin.php file at the following bit:

/**
 * Cross posting
 *
 * Possible values:
 *   - -1 if you don't want to use categories at all
 *   - 0 if you want users to post to a single category only
 *   - 1 if you want to be able to cross-post among multiple categories
 *   - 2 if you want to be able to cross-post among multiple blogs/categories
 *   - 3 if you want to be able to change main cat among blogs (which will move the
 *       posts from one blog to another; use with caution)
 *
 * @global int $allow_cross_posting
 */
$allow_cross_posting = 1;

So let's pretend you have 10 blogs (including blog #1) and you have a category called "dogs" in 5 of them. You change this parameter to "2" so you can cross-post across multiple blogs, then write something about dogs. In your back office sidebar you will see ALL the cats in ALL your blogs, and be able to check all your "dogs" categories as additional cats for this post. That means that the post will show up in each of the blogs that has a "dogs" cat or sub-cat, and - of course - in blog #1.

It's not the best imaginable method in the world, but it's what we've got to work with so there you go!

BTW the ability to have "one cat be in all blogs" has been brought up before. I don't think anyone has done it though. Being able to say "this cat is available in these blogs" would be even harder than the hack that's never been done.

#4: yeah you can do that. I don't know how, but it's really really REALLY easy. Someone else will probably pop in and say "do this" and that'll be that.