User registration, Member Login

1.)
I have several wordpress-driven websites hosted on my own server.
I want that users register on my first website only.
When the member is logged in, then the member shall be logged into all my other wordpress-driven websites without having to register or login on each of those other websites again. How can I achieve that?

2.)
a.) When a user registers on my wordpress-driven website, then the user shall enter more than just a username, password and email address. Is there a plugin that I can install and where I can create my own membership registration form for users who want to register on my website?
b.) Some wordpress-driven websites have a different CSS on the registration page. Where can I change the look of the registration page?

3.)
a.) Some wordpress-driven websites have the two login fields (username and password) on the main site. Is there a plugin or do I have to modify some codes from WordPress to achieve that on my website as well?
b.) I have seen (non-wordpress-driven) websites where the two login fields display the “welcome” greeting and the username. Is there a plugin that can do the same to my wordpress-driven website?

4.)
a.) In the navigation bar on top of the wordpress-driven website, usually all Pages are displayed. I want to make one single Page look like an own website by displaying the name of that single Page in the header of the website as a hyperlinked title and in the navigation bar its sub-pages (and sub-sub-pages and so forth of that single page) is displayed. How can I achieve that?
b.) For each Page (I don’t mean sub-pages) I have created a category with sub-categories where all posts of that single Page are contained. I want that the Categories widget only displays the categories of that particular page (the page whose name is displayed in the website’s header section as a title) and these categories should be displayed only on sub-pages of that Page. Currently, all categories of my whole website is displayed and I don’t like it because when a user is on that particular Page, then the user is only interested on categories of that single page only and not on categories of all Pages, that means on my whole website. So I want to connect a Category with one Page so that only the category and sub-categories for that page is displayed on that particular Page and its sub-pages. This should be possible easily with a plugin. Does anyone know such a plugin or can I change the wordpress file to do that and how?
c.) Posts display the date in the URL. When I edit a post, the date in the URL and in the post stays the same and it confuses my members. Is there a way to have the date to be updated whenever I edit the post? is there a way to update all posts I have created so far so that they all carry the current date?

5.)
a.) Is there a limit of pages and blog posts per one wordpress engine?
b.) Is there a limit how many members can write a comment at the same time?
If yes, what is the solution for the wordpress-driven website then?
c.) WashingtonPost.com is running on WordPress engine. Can anyone modify the core WordPress engine that we download from WordPress.org in order to make it capable of handling such a big news magazine like WashingtonPost.com?

Thanks in advance for detailed responses. I am new to WordPress.