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Shading would be good or significant amounts of water would do as well. For early growth, I would recommend a balanced fertilizer once every month or two.

It could be some form of Lilac bush. Maybe a Lochinch Butterfly Bush? There is also an evergreen type shrub named Munstead Lavender that has fragrant lavender blooms. Do you have a photo of it to share? That would help alot in the identification of it.

Stan

Here is an article that you may find helpful: https://www.gardeningknowhow.com/ornamental/shrubs/lilac/pruning-lilac-bushes-when-to-trim-lilac-bushes.htm

It may need some rejuvenation pruning. I would cut it back by 1/3 to a half. Also, give it some phosphorus heavy fertilizer. This will help with the blooms.

This article will help with pruning:
https://www.gardeningknowhow.com/ornamental/shrubs/lilac/pruning-lilac-bushes-when-to-trim-lilac-bushes.htm

It is probably a lack of phosphorus. Add a phosphorus rich fertilizer (normally sold as a "bloom boosting" fertilizer) to the soil. This article will help: https://www.gardeningknowhow.com/ornamental/shrubs/lilac/lilac-bush-not-blooming.htm

You can, but make sure you get as much of the root system and surrounding soil as possible.

The following articles should be of some help to you: https://www.gardeningknowhow.com/ornamental/shrubs/lilac/lilac-care-growing-and-planting-lilac-bush-plants.htm
https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm