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There are organic ways to kill the vine, but it sounds like you have ivy and these, even after they are dead, must be manually pulled off the building because of the type of roots they have. To kill the vine, you can try spraying it with bleach or pouring boiling water on it. Be advised that these methods will kill whatever the plant it comes in contact with.

It should be fine. As long as it was only exposed to rain and sun, it is still perfectly healthy.

The following articles will help: : https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm
https://www.gardeningknowhow.com/plant-problems/pests/homemade-aphid-control-a-natural-way-to-kill-aphids.htm

This article should help: https://www.gardeningknowhow.com/special/organic/homemade-pet-friendly-weed-killer.htm

You can try mulch, such as a straw mulch or sheets of newspaper laid over where you don't want weeds to grow.

You may find this article helpful:
https://www.gardeningknowhow.com/plant-problems/weeds/how-to-kill-grass-naturally-kill-unwanted-grass-in-your-yard.htm

I don't know of a quick solution, but applying mulch over the soil, particularly cheap mulch, will help to reduce the nitrogen over time. This is normally considered an issue, but for you it would be helpful. Squash plants will also help use up nitrogen in the soil. They won't produce much fruit though in this soil. Here is more information: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/too-much-nitrogen-soil.htm

You can also try adding a little phosphorus to balance the soil some, but I can't guarantee this would work. Some people also recommend gypsum.