drainage
I just bought a new house, and the drainage culvert out front along the road has several areas where the slope has collapsed, resulting in what looks like a miniature landslide landscape. What is the best way to shore this up to improve the aesthetics without causing drainage problems? 
This is a tough one. There is no easy way to remedy your problem. The best way would be to remove and replace the tiles properly. If in fact the tiles are real marble, you can grind them to create the proper angle and slowly refinish the surface by stepping down grits and finally getting to a wet polish grit of 2400. That is actually impracticable for a DIY if you don't have the equipment and materials to do it. If it were me, I'd remove a couple of tiles and reset them to fix your drainage problem. A couple of hacks with a hammer, scrape off some of the thinset, replace the tiles with a little premix thinset, regrout, wait a couple of days, seal the grout and your done. The actual labor time may be only a couple of hours, even though you have to wait a bit to use the shower again.
As others have suggested, a good excavator or even a landscape contractor might be able to find relatively simple ways to alter the landscape of your yard so that water will naturally flow away from the home.
Civil engineer would be your best bet but they are also expensive enough that I see them as the last ditch effort if other measures have been failing. Typically experienced landscape contractors or excavators can figure out where the water is flowing during a rainstorm and divert it away from the house with a bit of digging.
While water may not be seeping into your foundation right now, having all of that water pool next to the house is inviting problems. Reagrdless of landscaping, it is a good idea to install french drains around the perimeter of your house if you know that rain water is inevitably flowing towards your home.
The following image is a typical installation. The drain itself is typically a trench running a couple of feet typically from the house and going down below the frost line. You want the drain pipe to be deep enough that it will be below the frost line so that water flowing inside the pipe will not freeze and crack it. The frost line differs by region, obviously colder climates have a deeper frost line.
The holes allow water to flow down or up into the pipe, while the pipe running at a slight downward slope will take the water and flow it away from the house in another direction. It typically sits on a thin bed of gravel and the trench is filled with gravel rather than soil as water will more easily flow down through to the drain pipe if the trench is filled with gravel rather than soil. Soil will also potentially clog the drain making it ineffective. For aesthetic reasons, sod can be placed over top or you can do something clever like make a rock garden.
If you are a masochist and thoroughly enjoy pain, then you can grab a shovel and dig the trench, but for those of us who are lazy and like excuses to play with complicated equipment, you can rent a backhoe for a day and turn a week long digging job into a 2 hour weekend job. When digging however, be extremely careful about the location of underground utilities:
Water lines
Gas lines
Sewer lines
Electrical conduit
Cable and telephone lines.
Make sure to identify where these come into the house and approximate their location and depth in relation to where you intend to dig your trench. If any of them are close then it is probably best to call a professional just to be safe.
Another possiblity could simply be that your gutter downspouts are draining too close to the house, or are draining into storm drains that have become clogged.
The following is an example of a downspout draining too close to the foundation.
Many times this can be fixed rather easily, by extending the downspout horizontally away from the house, or channelling it far enough away that it has somewhere to flow other than down your foundation walls. Check out your local hardware store because there is a number of ways to effectively and cheaply handle this.
Also it is a good idea to check your gutters regularly for clogs. A clog can prevent water from flowing from your gutters to your downspouts, causing it to pour over the edge or sometimes down the side of the house. This is also something to check for.
Downspouts can also sometimes flow into an underground drain, which can sometimes be the source of the problem. This drain typically will be a storm drain that flows away from the house or to a public storm drain. These can sometimes become clogged causing water to not flow properly away from the house. Most landscape contractors can easily help unclog such drains if you suspect that this might be the case.
This should give you some ideas of things to look into or try before contacting a civil engineer, as the fix might be more manageable and less complicated than you think. And as far as a sinkhole swallowing your house, I have only ever heard of that happening because of a house sitting on top of a collapsed coal mine. If you live in a heavily mined area then your municipality probably has maps that will show whether your house sits overtop of an old mine. I wouldn't personally worry about this. The cracks in your foundation might be from the ground settling or moving slightly from excessive soil saturation.
Best Answer
Your pictures appear to show a steeply sloped section with erosion, a flat section, and another sloped section (without erosion.) If the flat section resulted from poor compaction of an original sloped profile that did not have the flat part, you may be able to simply fill that back in so the slope is not as steep. If you want to keep it in mown grass, that might be the best bet.
Otherwise, "rip-rap" or fairly large rocks are arguably the standard method of holding a steep drainage slope.