I was wondering if it would be safe to keep any chicks with mama in the main coop with everybody.
For thousands of years the way this was handled on most farms and other places was that the hen would hatch with the flock and raise the chicks with the flock from Day 1. You are dealing with living animals so you don't get guarantees, anything could possibly happen. But problems from other chickens were pretty rare, not often enough to change this model. I grew up on one of those farms and we never had a problem with the other chickens.
There is a huge difference in that model and what most of us are dealing with on here. Those flocks pretty much free ranged. They had unlimited room so Mama had plenty of room to separate her chicks from the rest of the flock during the day. During the good weather months, which was when the hen would be raising the chicks, they could pretty much feed themselves by foraging. They did not worry about percent protein in the feed, the chickens could forage for all they needed, usually without any supplemental food from humans. Most of us don't raise them that way. We keep them confined to coops and runs where Mama can't get good separation from the rest of the flock. They have to share feeders.
I don't know how big your area is. You are probably supplying all of their food so it could help to have separated feeders. I don't know how risky it would be for you to let a broody hen raise them from Day 1 with the flock. We all have different conditions. That means we get different results, not just with a broody hen hatching and raising chicks but with most things to do with chickens and their behaviors.
The chicks are at some risk from the other hens if they get separated from Mama where she can't protect them. I seldom have any other hen aggressively go after a chick but it can happen, especially if that chick invades the hen's personal space. To me, one of the big risks of isolating a broody and her chicks is when a chick escapes that confinement and is at risk from the rest of the flock. Mama can't get out to protect it. Some people say they have had broody hens that would not protect her chicks. I've never had a broody like that, but I believe them when they say that. Perhaps my 3,000 square feet in grass gives my broody hens the room to avoid those situations to start with.
I have two 5 week old chicks. when should I start introducing them into the flock? Should I even bring them outside?
Again, room can be critical to this. How big, in feet or meters, are your coop and your run? Please don't use words like "big" or "a lot", those don't mean anything. There are several threads on here where someone has a problem and say they have a lot of room, where the real problem is that they don't have nearly enough. Photos showing your coop and run and how they are arranged can help with specific suggestions.
Many of us believe that the younger you can introduce chicks to the flock the better. My brooder is in the coop so the chicks pretty much grow up with the flock. I can turn them loose at 5 weeks and they make their way with the rest. I've had broody hens wean their chicks at 3 weeks, leaving them on their own to make their way with the flock. Remember my 3,000 square feet I have outside. That makes a huge difference. And I have an 8' x 12' coop and two 4' x 8' shelters where they can sleep. The 8x12 coop is usually enough but a few times one of those shelters has come in handy.
One way chickens have learned to live together as a flock is that when there is conflict the weaker runs away from the stronger and avoids them. If they don't have enough room they can't do that. Until they mature enough to join the pecking order the chicks are lower in the pecking order than the more mature and are likely to get pecked if they invade their personal space. It usually doesn't take them long to learn to avoid the older ones. They can act scared to death of them because they are.
Dad would brood chicks in a cardboard box on the back porch and integrate them by taking them to the henhouse and turning them loose. It worked. He had unlimited room.
Many people on the forum successfully integrate young chicks all the time, even with limited room. Some of the generic tricks we use are to house them across wire for a while so they get used to each other, provide as much room as we can, improve the quality of what room we have by adding clutter, provide separate feed and water stations far apart so they can eat and drink without others seeing them, and trying to avoid locking them together in tight spaces. These are for all integrations, not just chicks. If size differences allow some use the "safe haven" method. That's where you provide openings big enough for the chicks but too small for the adults so the chicks can run to safety if they feel the need. Since my juveniles tend to not roost with the adults, I put up an additional roost lower than the adult roosts, separated a few feet horizontally from the adult roosts, and higher than the nests to give them a safe place to go that is not my nests.
I don't know what your facilities look like so I don't know which of these I might suggest or what other ideas some people might come up with. I may make it sound hopeless but it is not. People do it all the time, even with limited room. For some of us it is just easier than for others.