Mulches can help prevent soil erosion by 1) slowing the fall of rain and spreading that out more, and 2) by keeping the wind from picking up soil particles and moving them around. Compost and other types of organic matter can be laid down on top of the soil and, eventually, the Soil Food Web will move that material into the soil for you. That mix will be better all around than you can by tilling the same material in with much less effort on our part, however, this is just not as fast as you can, better just not as fast. Soil warming may be somewhat dependant on what soil type you have. My sand warms in the spring with no problem even when covered with a thick mulch layer. I have seen no problem with plants growing when they are supposed to and some people I correspond with around the world report the same experience while a few do say they need to pull back the mulch to warm the soil. But if you have a planting bed covered with mulch and you pull back a 6 inch wide swath to plant in does logic tell you that the soil in that bed will really warm up enough to encourage plant growth? If you removed all of the mulch that probably would happen and the soil moisture would be evaporated faster drying the soil out more. As a rule organic matter in soil will alter the structure so that in clay type soils water will move more freely while in sandy soils water will be held in the soil longer, but this has no affect on your water table. The water table is where the water flowing through your soil is at various times during the year and in the snow belt that rises, normally, in the spring and then falls during the dryer summer. There is always water flowing under your feet, even in the desert, even though it may be many feet down. That water table, the aquifer, is where those of us with wells get our water from. Adding organic matter to your soil will have no affect on where your water table is at any given time, that is determined by how much water is flowing through your soil from all around you. Oats and many other grains are reputed to have allelopathic properties (growth suppressing), while they are growing but may not keep that property once cut down, although some seem to do just that for a time. Some research seems to indicate that the more organic matter in the soil the stronger this property is.

Another vote for the PVC. Ask your landscapers for suggestions on how to handle the issues you are concerned about at the drop off. I can think of about three different methods to deal with that, off the top of my head, but one would need to actually see the site to fix on the best method. The best method is probably as Sofaspud mentions, terminate into a dry well. Once you trench, slope and install the PVC you are done. Any other solution presents ongoing maintenance issues of varying degrees. Whatever you do, do install a clean out somewhere close to the house so you can "jet" your hose into it periodically to wash out debris and critter nests.

Lots of inspiration links below via Google search for "gardening on a slope" and "hillside garden": My favorites are the mixed grasses and perennials. With the two it's fairly simple to combine a lot of plants that flower at different times to keep some seasonal interest. It would make for a very big ongoing show with low maintenance. http://www.gardeningknowhow.com/edible/vegetables/vgen/growing-a-vegtable-garden-on-a-hill.htm https://www.youtube.com/watch?v=X5xZgiibypc http://www.finegardening.com/shrubs-slopes http://www.doublescribble.net/images/3.lushome.com/wp-content/uploads/2013/05/garden-design-ideas-container-gardening-20.jpg https://s-media-cache-ak0.pinimg.com/736x/8b/f4/ed/8bf4ed7f15d135ea8568620f5a3506e5.jpg http://beautygarden.website/wp-content/uploads/2015/09/small-garden-design-sydney-garden-design-seaforth-pictures.jpg http://www.livingandlearningwithluisa.com/2013/05/11/building-garden-beds-on-a-slope/ http://decotrack.info/wp-content/uploads/2016/03/beautiful-hillside-garden-patio-landscape-ideas-best-design-281142.jpg http://www.santacruz.watersavingplants.com/GardenWorks/Photos/851.jpg https://s-media-cache-ak0.pinimg.com/236x/9a/73/01/9a730133c0f456984a6fe9c404344f4b.jpg http://www.sunset.com/garden/landscaping-design/hillside-gardens/how-to-landscape-a-slope https://www.pinterest.com/explore/hillside-garden/

I would create very shallow swales across the slope, so the water runs a diagonal down the slope rather then straight down. Since the area that is being drained, is relatively small and you do have gutters, the swales need to be only about a 2" deep and about 18 to 24 inches wide. The swales do not need to be soo deep they are a problem when mowing the yard. They will slow the water flow, and stop the erosion. If this area is in the shade most of the day, the Mulch bed seems like a good idea. If it is in sun at least part of the day I would go for grass after preparing the swale. As for grass it depends on where you are, and the actual amount of sun the slope received.