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Your 2 big challenges are going to be preventing further erosion and during soil prep and while your new plants are getting established. There are erosion control blankets and grids that are pressed into the soil. (search for images of erosion control grids) Perennial weeds will have dropped seed into the soil. You can choose herbicide to kill present weeds, water and let the seeds sprout and spray again; you can hand pull; and you can cover the area with a plastic tarp for the summer which will kill weeds and seeds. So there is no easy answer except to build a retaining wall and forget the slope! Search online for groundcovers for Georgia and figure out how many hours of direct sunlight the area gets. Here is an article on groundcovers for Georgia. https://secure.caes.uga.edu/extension/publications/files/pdf/C%20928_2.PDF

The only risks in trying this are ending up with 2 different turf grasses, 2 different growth rates and color. If a perfect lawn isn't your thing, then give it a try.

This could be used for compost tea, but I would suspect that this is to water a lawn without a sprinkler system. Can you include photos, so that I may see what is going on?

This website will be the closest I could get you to watering schedules for your climate zone: https://www.tripsavvy.com/summer-lawns-watering-tips-and-schedule-2681592

How wonderful for you to take over your family's garden.

The only flower that can be seen and identified from your images are the low purple blooming plants. These are in the Lungwort family.

https://www.gardeningknowhow.com/ornamental/flowers/lungwort/growing-lungwort-flower.htm

You may need to wait for the plants to bloom to make id's. You can flag plants that you want to keep or remove.

It can be fine, as long as the ground is dry. There will be no need to water if the rains are keeping the ground moist. Otherwise, it will be ok to water during dry spells.