As for the rooster, at six weeks you should be fine. Most roosters consider any chicks to be his. Sometimes mine help broody hens take care of the chicks but most of the time he just ignores them. When I integrate 5 week old or 8 week old brooder-raised chicks he ignores them but the brooder and grow-out pen is where he can see them so they kind of grow up with him. If you wait until they hit puberty and he sees them as rivals that can change. You don't get guarantees with living animals and their behaviors but your biggest threat is much more likely to come from the hens than the rooster. I seldom have that problem with hens either but I have a lot of room.
I have no idea of how much room you have in the coop or outside. I don't know how your coop is set up. I don't know how many chickens you have. It's hard for me to give specific suggestions if I don't know what you have to work with.
In generic terms, I'd suggest you house those chicks somewhere that the flock can see them for a couple of weeks before you let them mingle. When you do, try to let them mingle on their terms. Don't try to force them into a small shared space if you can avoid it. I like for them to have a separate place to sleep until they want to move up with the adults. Separate and well-spaced out feeding and watering stations help.
