Mosquitos will live where there is water and CO2. Remove water and rotting debris and they will not hang around.

Your question focuses on the shed, but I think this is an error of design. Focus on your garden as a whole instead --- where you will walk, how the garden will be used -- and your design will make more sense.

Gardens that support animal life are not tidy -- flower heads left to go to seed for birds, etc. Be sure that is what you want before you let the wildness in. Imagine a wild meadow with paths through it; that would be the look.

As to the garage wall, I would remove the sloppy ivy and put some tall ferns back there and/or the least aggressive vine you can grow. Be very careful with vine selection, they can become monsters over time.

This looks like one of those situations where you will really need to be designing and maintaining the back of your neighbor's property. Not 100% sure, but I think you are trying to screen the concrete block wall and it belongs to your neighbor ...? It also looks like the English ivy emanates from his property ....? If that's the case, you won't be able to get rid of it. It also looks like there is another vine -- honeysuckle perhaps -- in the mix. The two vines mixing together are not going to be easily maintained or look good. I would get rid of ( or limit) the honeysuckle. I'd allow the English ivy to grow on the block wall to the extent that it covered it completely. Just keep it off the wood parts up at the roof. You'd only need to clip the face of the ivy periodically to keep it smooth and trim the edges at the roof. You could trim mechanically or with herbicide (if careful.) You would also need to trim it at the ground to keep the ivy from spreading into your yard. Actually, you could limit the ivy to the space between the buildings and just keep it off of your building and from encroaching into your yard. There is so little room between the structures that it is barely worth planting anything additional. But you could plant Hosta or some such thing if you like. I would not give you a half-penny for the theory that garlic will keep mosquitoes away. It would probably not grow well there either if there is substantial shade.