You probably don't want grass anyway. If it's really shady, it won't grow well. And if it grows well, well then, you have to cut the darn stuff!

I'm not sure I understand the topography; the yard slopes up from your patio? Or is simply that there is a depression five feet out from the patio where the water flows past? If the yard slopes up from the patio -- that's a building problem and the landlord needs to work on that.

For the downspout right beside the door, you might want to dig a trench out several feet from the door, maybe 7-8" deep and wide, and fill it halfway with gravel, then dirt on top of that -- just to help carry the water away from the surface.

You might also want to get a downspout diverter. {{gwi:1257319}} They are really cheap, and they can also help carry the water away from the door.

As far as plants -- think lush, rainforest tropical. It's not all green, either; coleus loves the shade but comes in all sorts of gaudy colors and patters. Try doing a google image serach on "shade garden plants" and look at the pictures that pop up. When you find one you like, do a little research on it (Dave's Garden has a wonderful database; be sure to check the users comments about growing the plants -- some great real-world info there.)

Here is a link that might be useful: Dave's Garden plant files

Thanks!
As for the water situation, I am the second-to-last apartment. There are 10 to the left of me as you look at the picture of the patio itself. The water from the building's first downspout actually runs away from the building, which starts the 'river' that's 5 feet out from the edge of the patio. I think water runs down from the hill where the trees are as well, merging to make the river even worse. Some water must run along the building, and about 3/4 of the water that comes out of my downspout actually goes directly into the river that encroaches on my patio. You can see in the picture where the previous tenant thought to put brick down to divert the river. Didn't work! The land behind my apartment actually slopes slightly downward to the right as you look at the patio. Being one of the last in the row, I get the water from almost the entire length of the building!

I may just leave the back dirt as it is and focus more on making the patio itself more enjoyable with a container garden. I'm thinking of putting two planters parallel to the house on the border of the patio. I'm thinking the kind that are up on stands about two-three feet off the ground. Or, I was considering buying two small sections of an iron gate from Lowes and putting them in the same place, but with window-box type planters. Basically, I would create a balcony for myself instead of a patio, leaving a small area between the two sections of gate to walk through in the center...

You have a cute usable space there! An idea would be to put down some patio blocks to walk on, with crushed stone or a mulch between them. Lowes has wooden paver blocks if you want that look. You would need to edge to perimeter so nothing washes away. Many times you can find free red bricks for edging (or use broken concrete pieces if allowable there). It depends upon your bedget and how much time you want to put into this project. You can add some raised planter boxes on the sides, maybe with lattice for privacy. Even large pots at the sides and back would look nice. There are many shade plants to use, with impatiens and hosta being the staples. Here are a few ideas (minus the shrubs and trees planted)...
{{gwi:1257320}}

What a wonderful idea! I've seen some of the townhouses in the complex extend their 'block' patios with cute stone or tile, but I hadn't thought it would work for me. Since I have no grass, it's not like a maintenance guy on a lawnmower would ever interfere with it!
I like the idea of the raised beds in the background too. Though I can find the raised 'container planters' for $25 per locally right now, a raised stone bed would allow me to extend out my patio, giving me the room I want. This is definitely something to propose to my friends in the office (to make SURE the lawnmower bandits don't run it over!). Thanks so much! I hadn't even considered extending it out. I was kind of sad that I would feel like I lost space by enclosing with flower boxes...grilling on my teeny charcoal grill is tight enough without a physical boundary to abide by.... :-)
I know I haven't mentioned it before because he only really walks out in the front yard and around the block, but I have a 10 month old boxer. If I could create raised beds, it would give me a foundation to put up a small baby-gate to let him out back with me. We're batteling allergies, so right now he's not allowed around any of the plants out back...hopefully we'll figure out what he's allergic to very soon so I can let him out there with me again! This might be the perfect solution...protect him from the allergens while increasing my back patio and his playroom...