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Rust is caused by a fungus and it will appear as more of an orange or rust color. Here is more information: https://www.gardeningknowhow.com/plant-problems/disease/learn-about-plant-rust-disease-and-rust-treatment.htm

It sounds like you have plenty of nitrogen in the soil but you may be lacking phosphorus. If there is a lack of phosphorus in the soil, plants will have little to no blooms. You can either use a phosphorus rich fertilizer or add some bone meal to the soil. These articles may help as well: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/hydrangea-not-blooming.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/phosphorus-plant-growth.htm

Actually, hydrangeas prefer shady situations as opposed to full sun, so planting them in your backyard should be fine. This article will help with caring for your plants: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/growing-hydrangeas-hydrangea-care-guide.htm

Try a high phosphorus fertilizer or add bone meal to the soil around your shrub and see if this helps. Phosphorus is responsible for flowering in plants and a lack of it in the soil will result in little to no blooming.

You may be lacking phosphorus. If there is a lack of phosphorus in the soil, plants will have little to no blooms. You can either use a phosphorus rich fertilizer or add some bone meal to the soil. These articles may help as well: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/hydrangea-not-blooming.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/phosphorus-plant-growth.htm