1. AKA header with "a horizontal menu" (fairly common)
2. I don't get what the "toilet roll" effect is, please explain. You can make your index page have a list of just the post titles, and clicking the title takes you to the full post. Or you could put the Title, and the "Excerpt" which is like the first 40 words of your post or so.
For linking some of the post-titles to external sites, and some to your internal post, I wouldn't know how to do (this is not common). What you would probably want to do is make the Post, and put your Link for external sites in the Body of the Post. The title might be something like "Cool site I found" and the body contains the link.
For just pure links, you can use the so-called Blogroll function of wordpress, it pretty much displays your links in the sidebar w/ whatever you want (URL, description of each link if you want).
3. I don't think it is a "child page", I think you mean a "Single Post page." Child Page is something different. Keep in mind WP has Posts, and Pages, which are similar but different. Most themes have a "Single Post Page" template which has the same header as the index, so that's good for what you want.
4. I don't know quite what you mean. Most themes have a left-justified Blogroll (aka Links list) in the sidebar. You can move it to an "inset" if you know the CSS to display it, there's just a simple "template tag" you can learn from the Codex to put your blogroll in whatever CSS div you wish.
5. Disable all the Comment options in the Admin. Every theme has Comments built in -- but you can easily delete that section of the theme, if you don't mind getting your hands dirty. This is not a difficult hack for a first-timer, just learn some basics first like: Always make a back-up of the theme file before you edit it. If you've got that, you can experiment with deleting the parts of code that refer to "comments". You'll figure it out, or ask on here.
6. Lots of themes have sidebars, some do not. Usually ones without a sidebar are called "1 column themes". Try searching on the themes site for 1 column themes. If you get a theme that has a sidebar it is possible to remove it, but some themes this would be easy, some themes difficult. With the Default built-in theme (aka Kubrick) it is very easy to remove the sidebar -- if you want to do that just search or ask how on here.
7. The RSS feed of the Articles list is built in. You already have it now. You just want to pick a theme that displays the RSS link to your readers so they know about it. But it's there, even without a link, right now. You actually have a bunch of feeds already -- Articles feed, Comments feed, Author feeds, heck you can even get a feed for a certain month / day / year. Atom Feeds, RSS 1 and RSS 2 as well. This is part of the beauty of WP.
Site search is built-in, you have it now, just put the Search template-tag in your theme template (the theme you choose probably already has this).
It is commonly recommended to NOT use the default Kubrick theme if you wish to modify it, because some things are tricky to modify. But I have found it is fairly easy. I would just use the Default theme if I was you. It has most of what you want, and the hacks wouldn't be too tough if you don't mind learning, searching, reading the Codex, and asking.
But other people would doubtless recommend that you go to the Themes site, and pick a 1-column theme that looks similar to what you want. Then spend the time to hack the index to only show what you want, and delete what you don't want.
Hope this helps