You are asking two questions that deal with "Mother Nature" so it is impossible to give you a definitive answer. You can check out monthly averages for temperatures & rainfalls for Antigua at www.weatherbase.com. Beyond that no one can predict specific weather for mid-June. Chances are you'll see daytime temps in the mid to high 80's F. with plenty of sun and blus skies dotted with white puffy clouds and you might encounter occasional brief and localized showers. Of course, if a tropical front happens to be passing through the Caribbean during the specific dates you will be on island the weather can be completely different. Unless you are in a position to cancel and re-schedule your trip you should worry about the weather. It will be what it will be and you can't do anything to change it.

As for sandflies, they are not particularly prevallant on the island but you might encounter mosquitoes and "no-see-um" if there happend to be some prolonged periods of rain in the week or so prior to your visit. You'll encounter them (if any) mostly in the evenig just after sunset and more on the beach than off-beach. If there's breezes the bug population diminishes.