Here is some input based on each picture.
1) See the drain pipe? Even when you fix the eroded part of the soil, the water rushing out that pipe will keep eroding it.
Here is what I would do:
Trench to a lower area, hopefully lower than your house, where it is safe to discharge water. Put in proper pipe (not that flexible stuff). Connect the down spouts and any area drains. Where the pipe comes out put a bed of gravel or cobbles (rounded river rock). You might make it look like a dry creek bed when there is no rain, with large boulders and so on. When it does rain the water will be spread out to soak in, not rushing across the soil.
2) See notes about pic 1 about not allowing water to rush across the soil.
To repair something like this I would rototill the surrounding area to generate soil to fill, and rake it smooth. Then add soil conditioner, compost or similar material to the area you have rototilled.
Reason: When you mix different types of soil they may not blend well, and plant roots can have a hard time growing from one type to another. Also, the watering needs of different types of soil are so completely different that you would have a very hard time keeping up with it.
By spreading out the existing soil, then blending it with compost or other materials in a more uniform way you are creating a better, more uniform area for the lawn or other planting.
If this is an area where water will continue to rush through and cause erosion then make something like described in note 1: A cobbled, dry creek effect works really well.
3) Rocks larger than 2": get them out of the lawn area. Rocks to about 1" can usually be rototilled in, and if they are still at the surface remove them. But deeper in the soil they are not generally a problem (unless there are a LOT). Between 1-2" depends on how many there are. A few = no problem. A lot = remove them.
4) The sand. How deep is it? Can you scatter it over a wider area so it is not pure sand right there, then add soil conditioner to the low area where the sand was and rototill it to build the soil high enough? If the area is so low that you cannot fix it with about 2" of soil conditioner, rototilled in, then you will have to bring in soil. Try to find soil that is as much like your existing soil as possible. Build up the area in layers and rototill the layers with the existing soil to create a transition zone. Leave it 2" low and add soil conditioner, and rototill that in. Net result is a transition from rich soil with lots of soil conditioner (about 1 part soil conditioner: 2 parts soil) down to a blend of local soil and imported soil that transitions to just the original soil.
See notes about photo 2, about not creating areas with different types of soil. An area that is high in sand will drain so fast it will need frequent water. An area with finer, denser soil will not need that frequent water, you would drown the plants if you watered enough to keep the sandy area moist. So... move the sand, do something else with it like a specialty garden of plants that need such fast drainage, or spread it around more or less equally throughout the garden.
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Best way to get a lawn going:
1) irrigation (if needed). Below grade pipes, sprinklers...
2) Prepare the soil. If you are not familiar with what your soil needs the best places to find out are local nurseries, department of agriculture, or university extension offices. I can tell you how to do a pretty basic soil test.
3) Rake the area level. There are many tricks to doing this on a large area. If you are just patching, then a rake will work just fine. Remember that the prepared areas will settle, especially if you rototilled them, so leave them gently mounded so that they will settle to the right level. You can then tamp these areas with a 2 x 4 on edge, and this will help to settle the soil without compacting it too much.
4) Scatter grass seed (ask those same experts for their recommendations for the best lawn see for your area).
5) Cover the seed with barely enough compost to hide it from the birds. Since you have taken care of any soil erosion problems you do not need to use straw.
6) Roll the area with a large drum (called a lawn roller) to tamp down the seed and make sure it is in firm contact with the soil. For small areas and patches you can simply walk on it to do this, or tamp it with the same 2 x 4 you used earlier.
7) Keep the area damp. Lawn seed takes about a week to show green sprouts. Depending on the weather you might have to lightly mist it several times a day if the weather is hot. In milder weather, or rainy weather you will not need to add much water.
As the grass gets going it will need deeper water, but not so often. Daily becomes every other day, then every 3-4 days... but when you water you will need to run the sprinklers longer to deep soak.
Avoid walking on the newly seeded areas until the grass is stronger. A month to 6 weeks is common. Then you can get out there and remove weeds. A good trick is to lay a board down over the new grass to distribute your weight.
