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There are two problems that you will face, unfortunately.

I'm not aware of any grass that would be able to outcompete ivy in the shade, since they don't tend to do as well there.

Ivy is an absolute nightmare to get rid of, sometimes. Replacing it will mean killing it and putting something more aggressive, or as aggressive, in its place.

This article will suggest shade ground covers that might suit the area:

https://www.gardeningknowhow.com/special/shade/ground-covers-for-shade.htm

I would cut the brown area back by about 2/3 to get rid of most of the issues. They should regrow quickly. The sooty mold was probably a result of aphids or scale. Continue to apply Neem oil till the insects are gone. Make sure the white spots aren't mealybugs. The narrow area where it is planted probably needs a boost of compost or manure. Make sure they are getting adequate water.

https://www.gardeningknowhow.com/plant-problems/pests/insects/mealybugs-control.htm
https://www.gardeningknowhow.com/ornamental/groundcover/english-ivy/english-ivy-plant-care.htm
https://www.gardeningknowhow.com/plant-problems/pests/insects/what-is-armored-scale.htm
https://www.gardeningknowhow.com/plant-problems/pests/insects/homemade-aphid-control.htm

This resembles damage from soils that remain too moist for too long, unfortunately.Fungicides will necessary, and if you can, try and let the soil dry out down to 1/3 of the way into the soil before letting it get watered again.

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

There are a number of causes that could lead to this. It could be bacterial, fungal, or viral.

Because it has been established with no issue for a long period, I am less likely to assume that this is to improper care.

Copper fungicides will be the best option in this case. With the exceptions of viral issues, it will take care of most microbial issues.

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

https://www.gardeningknowhow.com/garden-how-to/info/what-is-copper-fungicide.htm

Your local extension service can, also, be of assistance:

https://www.gardeningknowhow.com/extension-search

It can mean a number of things, but you can keep a close eye on these parameters.

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-moisture-in-plants.htm

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-soil.htm

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soil-ph-plants.htm

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/test-soil-for-pests-and-diseases.htm

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/making-sure-soil-drains-well.htm

It is very hard to say what is going on, here. If this is a fresh planting, then it could be that it is just growing new growth. If this is in shade, then it could be lacking, slightly, in light. Another possibility could be nitrogen deficiency, or pH imbalances. It could, also, be soils that remain too moist for too long. Unfortunately, without checking for certain things, it can be hard to say.

In the meantime, this article will help you to care for the plant:

https://www.gardeningknowhow.com/houseplants/ivy/variegated-ivy-care.htm

Unfortunately, your photos did not come through. Usually, though, a little bark damage will not harm the tree. Just care for it as it should be, and it should fare just fine.

https://www.gardeningknowhow.com/ornamental/shrubs/elaeagnus