Okay, so this is only like a 5' by 10' area?
#1: More than a couple inches of top soil will likely smother the old grass and weeds, but you may still get some seeds into the new dirt, though that can happen even with chemicals.
#2: You can pick up a bottle of roundup (glyphosate) and spray, but then you really should give it a good week+ before planting in that area.
#3:This works best if it is hot out, as then it is extra hard on the grass and weeds. Set your mower to the lowest setting. Mow everything you want gone. You can leave it sit for a couple days and watch your lawn die or just start putting dirt over it. Unless you have some really aggressive weeds, this should do just fine.
After new dirt / soil is down, mix a generous amount of peat moss in the new soil, down to 6" if there is that much new soil. Then plant grass seed; top dress with peat to keep the seed moist. Water generously, but do not flood.
If there is a tree in this area, please post photos, as that will need to be addressed, do not simply fill dirt in around the tree.
Re: washing away; You can use some annual grass as well as regular grass seed, the annual will take hold much faster, within a couple days if the weather is warm and the area is well prepared and watered. Hay (ensure it is weed free) or straw put over the area may help. Without a photo, it is hard to say. Use some temporary edging if you feel it is needed.
