Hi @mtaylor,
I hope you are well today and thank you for your questions.
1. Add a new div class on the static front page between the header and post-inner-content 2. Have the slider appear in the new div class below the header
To achieve this you have to create child theme of Sparkling theme and develop custom code in it.
You can refer the child theme solutions shared in the following reply to develop it.
https://colorlibsupport.com/t/no-featured-image-from-blog-post-on-the-flexslider/
3. Remove the auto timing transition on the slider photo change
You can achieve this by using the solution posted in the following reply.
https://colorlibsupport.com/t/slider-timer/#post-42590
4. Style the slider transition buttons individually
Could you please tell me how you want it to style so that i can help you?
Best Regards,
Movin