No. That is the one hard and fast rule: Never feed layer feed to a chick under 18 weeks of age. The reason is that the extra calcium will damage organs, possibly even causing death.
After that, you can feed chick starter or grower to laying hens, you can feed most anything to a rooster, or you can feed your whole flock a flock raiser feed. If you feed laying hens something that does not have extra calcium (layer feed,) you may want to put out a dish of oyster shell separately, for those hens who feel the need. Chicks will generally pay little or no attention to oyster shell. For adults, it is probaby healthier to offer something other than layer to those who are not laying, such as your roosters, molting birds and old hens.
Many people do not offer grit, and their flocks do fine, because they find enough small hard rocks while foraging to grind their food in their crop. To be safe, you might want to offer this separately. The exception is, chick starter has enough grit mixed in for the chicks.