OK, let's take these in the order you presented them.
1. OK, your permalinks are fine. Personally, I would take the date out of them because it's supposed to be better for SEO, but if you want the date in them, then you can leave it. (I would still put the date at the end, however, not before the title. Again, it may help slightly with SEO.)
2. Yes, that's your ACTUAL homepage. It seems fine.
3. That is just a page you made. You can delete it in your Admin area by going to Page > All Pages. Then just find it and delete it.
In order to get your menu that says "Home" to go to your actual homepage (http://www.thelyonsshare.org), then you'll just need to change your menu. In your Admin area, go to Appearance > Menus. You can change it there.
4. I'm not exactly sure what you want here. When you're on your actual homepage (http://thelyonsshare.org/), and you click on a title, it goes to the full length version of that single post.
You seem to be asking for a page that will show all your full posts one after another -- i.e. like your homepage but with full posts instead of excerpts. Is that right?
If you'd like, you can set up your actual homepage to show full posts instead of excerpts. Personally, I would keep the excerpts because they're easier to browse. Also, some seem to think it helps for SEO purposes because you don't have your full post showing in two different places (this is called duplicate content and search engines don't like it). Google should be able to understand what you're doing, and they say they do, but what Google says it does is not always what it does in reality.
Anyway, it can be done by digging into the code a little and changing one thing. In your Admin area, go to Appearance > Editor > index.php. You'll need to click the index.php file on the right hand side (also probably called Main Index Template) so that you can edit it.
Once you have that file open in the editor, search for the_excerpt and replace it with the_content. Save the changes, and that should do it.
Note - This will only do it for that particular theme. Also, if your theme is somewhat complex, then you may not find the_excerpt in the index.php file. If that's the case, then it's in another file that is being called into the index.php file.
Hope this helps.