What month is the best time to cut it back to about four feet? Can it be cut back in spring before it gets leaves, or must I wait?
Wait till after it has finished blooming to prune. If you prune before then, you will lose some of the flowers. Other than flower loss, it will not hurt to prune earlier, but most people like to wait until after they have enjoyed the flower. This article will help:
Please suggest a spring fertilizer for my lilac bush. I live in the desert and summer temperatures can be over 100 degrees for several months. I am 'babying' this little bush (3' high) but am fearful the searing sun may do it damage. Should I be shading it in some manner during the hottest months?
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.
I have a lovely smelling bush with tiny clumps of light lavender flowers. It is so fragrant, and a friend wants to know its name. It has been here for 12 years and I cannot remember. Is there any way you can help me? The clusters of flowers are no bigger than an inch or more in width.
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.
I have dead flowers from last year's spring bloom. Can I cut them off for this spring? One year I did that and I did not get any blooms. Help! I go through this every year and love my Lilac tree (it's like 15 years old and 9 feet tall).
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
We recently moved to a new house that has a lilac bush in the front yard; however, it is very thin looking and the flowers on it are very small. It is rather large though, standing about 7 feet tall and about 5 feet wide. Is there something that I can do to encourage stronger growth and flowers (it looks like it could use some steroids)?
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:
I have two French lilac bushes that haven't bloomed since they were planted three years ago. They are small (5 feet) and sparse, but look otherwise healthy. They do receive sun and I see no bugs. I was told sometime ago to try hitting the trunk with a shovel to 'shock' it into blooming, but this did nothing. Any ideas for me?
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
Can you transplant a lilac tree? It is about six to seven feet tall. If so, please advise on the specifics.
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