Dynamic Content Grid on Homepage

First, here are the sites at hand:
The site I’m working on. It’s twenty-eleven theme with extensive modifications in a child theme. It’s part of a three language network that I’m not finished working on yet.
Amnesty International’s site, which I’m roughly modeling my site after.

Okay, so what I’m trying to do is get a grid of a few content boxes beneath the content gallery at the top of the main page. Each box will have a category it corresponds to, and it displays the most recent post from only that category. That way, every time I make a new post in “E-publications” it automatically becomes the new piece of content in the “E-publications” box.

See how Amnesty has the ‘News’, ‘In Focus’, ‘Livewire’ boxes? I would like something similar to that. Maybe 5 boxes. The non-profit I volunteer at has several publications. I would like the top two boxes each to show a link and an excerpt to the most recent of the corresponding category. It’d be great to have a box dedicated to videos too.

I’m open to using a plug-in, and I have tried several out already, but I can’t find any that do what I’m trying to do. The Dynamic Content Gallery (which I highly recommend) I’m using does exactly this thing. I can even specify which categories to include/exclude and how many posts to skip. I would love that same functionality with what I’m doing.

If I can get this to work, I’ll probably make the home page so that no posts are shown at all (a static page, I guess), just the content gallery, the grid I’m working for, the sidebar, and the footer I have yet to build.

So, can I do this without changing themes? Are there any plugins that are designed for this purpose, and allow each box to have its own category?