I leave the rug on but not the mask - i find the horses have trouble enough seeing through those things as it is, let alone when its pitch black in the middle of the night!
Fly rug yes (he is a sensitive soul, flies love eating him), mask sometimes depending on how hot it was and what time I left the yard. Oh this is only on the cob, the Fjord doesnt even own a fly rug.
It depends what type of bugs - for those who get sweetitch, midges have a morning biting session between 5am and 7am depending on sunrise, so if the horse isn't protected at that time, he will get bitten. I was amazed that people think midges are only around in the evening, I've been out and about plenty of times early in the morning and bitten by the critters, so the horses are too!
I leave the both on, i don't really like leaving the masks on especially as one of mine is black but the flies are so bad here until it's dark basically so i'm leaving them on at the mo
I used to take Suni's mask off at night until last weekend when I went down in the morning and she had an eye infection from the flies. Her eye were all sore, puffy and gooey so now she comes in in the day and out at night with mask on. Even though I'd prefer her not to have too.