Whether or not full posts, permalinked post titles or thumbnails are displayed on Categories Pages and Archives pages is theme dependent ie. it's coded into the template and we cannot edit templates or themes. http://en.support.wordpress.com/themes/editing-themes/
In the majority of themes, posts on category or other archive pages show up just as they do on the main posts page in the blog and the number displayed is also the same. (See this page > Settings > Reading and note how many posts you choose there sets it for all). But in some themes only the beginnings of posts are shown, and in a few themes you only get titles or thumbnails. See here for a theme list of themes that may display the way you want > Category/archive pages: full posts or not http://wpbtips.wordpress.com/2009/03/25/full-posts-in-archive-pages/
2. I cannot post a picture with text when I choose a category to post in. The picture only shows on my home page or if I click on the date under the text from the post in the category.
If the theme has an active excerpt box below the post editor, you can control the length of the excerpt displayed and insert a thumbnail image.
You compose your whole post first and before you publish it you switch to the HTML (Text) editor, copy the code you want in the excerpt, paste it into the Excerpt box, and publish the post. http://en.support.wordpress.com/splitting-content/excerpts/#creating-excerpts
This will have to be done on every post.