First, the fairly easy one. Instead of feeding both Layer and Grower, when I have mixed age flocks, I feed them all Grower with Oyster Shell on the side. The ones that need the extra calcium for the egg shells can get it while the ones that don't need it don't eat enough to damage themselves.
Will Layer hurt 14 week old chicks? Probably not, or not much. There are plenty of studies that show that feeding Layer to chicks can cause serious damage to the chicks. These studies always start off feeding nothing but Layer to day old chicks. I have not seen a study that looks at starting to feed Layer at 8, 10, or 14 weeks, so I don't know when the cut-off date for damage is. And it will be important what else they are eating. It is not the percentaage of calcium in the feed that matters, but how much total calcium they take in. If they are eating a lot of other stuff besides the Layer, they may not eat enough calcium to harm themselves. And the effects of the extra calcium is not due to just one time of eating a lot of calcium. The damage is going to come from a continued exposure to extra calcium. A lot of times the damage is internal, if it is there. In these studies, some chicks die because of the effects of the calcium, but they have to look at the internal organs to see a lot of the damage. It is not always evident.
I can't tell you if feeding Layer will harm your chicks. There are just too many unknowns. But I just feed Grower with oyster shells on the side until they are ready to lay so I don't have to worry about it.
Now the hard one, introducing new chickens. A broody hen may wean her chicks as early as 4 weeks of age. These chicks are fully integrated with the flock, but since they are so low in the pecking order, they get picked on a lot. If they have enough room to get away from the older chickens, they do quite well, but they will keep their distance to keep from getting pecked on. I have no real problems integrating 8 week old brooder raised chicks with my flock, but I have a lot of room and I house them side by side from Day 1. My brooder is in the coop.
It helps tremendously to house them where they can see each other for a while, at least a week. Chickens can be territorial and defend their territory from intruders. It does not always happen, it depends a lot on the individual personality of your chickens, but when you get a territorial hen, it can get bad. If they are used to seeing other chickens in the area, they are less likely to go after them as intruders. Or if you introduce them in strange territory where the older flock does not have a territorial attachment, it can help, but I really like housing them side by side for a while.
You still have the pecking order issue. It is perfectly acceptable in chicken society for a chicken higher in the pecking order to peck a lower ranked chicken if her personal space is invaded. It does not happen every time two chickens get close together, but if the higher ranked one feels challenged at all, she is expected to defend her social position or get knocked down. So she pecks. If the lower ranked one runs away, all is good. But if it does not run away, that is a challenge to her priviledged position and it can get pretty rough.
Since a more mature chicken will automatically outrank a less mature chicken, you need to give the younger ones a chance to run away. Space is a big help, but not all of us have unlimited space. Give them as much as you can. A lot of times the lower ranked chickens spend a lot of time on the roosts, up out of the way, so plenty of roost space or an extra perch in the run might help. Mine also hide a lot. Try to give them things to hide under or behind. Some way to avoid the older ones.
I find that providing extra feeding and drinking stations helps reduce areas of conflict. Sometimes older chickens will not allow the younger to eat or drink. It's part of the intimidation process.
The worst times for mine are on the roosts as they are settling in for bed. If the roost space is a bit limited, they cannot get away from a bully and stay on the roosts. This will sometimes cause the younger ones to try to find a safer place to sleep, maybe in the nesting boxes or even outside the coop. I put up some roosts away from the main ones and a little lower down so the young ones can get away.
Sometimes integration goes so smoothly you wonder what all the fuss and concern was all about. Sometimes it gets really bad. The personality of your individual chickens and your circumstances and set-up is different than mine or anyone elses. I can't tell you how it will go with yours, but I wish you good luck when you try it.