There's no way to know for sure if a cat is actually in pain while in heat but I believe they are since when a cat is in heat they Meow and whine like they're in pain.
So in my opinion and from what I've seen cats do when they're in heat I would say that yes that cats experience pain while in heat and cats can be in heat as early as 4 and /12 months of age so it's good to get the cat spayed before the cat reaches 6 months of age.