Hi,
I’m sorry I couldn’t answer, my inbox was full of queries I needed to answer first.
1. Styling a drop-down is sort of limited to browsers, ajax search pro does not use any custom styles, it falls back to the browser defaults. I recommend using custom CSS for this. Some aspects can be changed, others may not. I recommend checking this and this article about possibilities.
2. You can by forcing a nowrap line break method for the container via custom CSS:
.asp_shortcodes_container {
flex-wrap: nowrap! important;
}3. This one is hard to tell, I may need back-end access to check the configuration first. It can be multiple things – like taxonomy or custom field exclusions, limits not set correctly etc..