Yep, you have to show Mr. Rooster who's boss (that's you).
One thing you might try if you're determined to keep him, is to put some kind of soundproofing in his coop and block light from coming in early in the morning. Roosters do crow anytime of day but if the neighbors don't like him waking them up and can tolerate his daytime noise, maybe that would be a compromise for everyone.
Never did make sense to me that folks that move out to the country don't like country noises (it sounds as if that's the situation you have). And barking dogs? Not for me! If a dog only barks when it's excited about an intruder or some similar issue, that's one thing...the ones around us do it all day long. I guess they're bored and/or badly trained. There are 18 dogs between the 3 houses closest to us. That's a lot more noise than an occasional rooster crow if you ask me (not that you did, lol).
Anyway, hope things work out for you.