No, you do not need a rooster for hens to lay. Commercial egg producers do not include roosters with the laying flock. The only reason you need a rooster is if you want fertile eggs. The rest is just personal preference.
That was easy. Now it gets a lot harder, when to butcher dual purpose cockerels. Some general information first. The older a chicken gets, especially a cockerel, the tougher and more flavorful the meat gets. The broilers you buy at the store are babies, 6 to 8 weeks old. They are especially tender and pretty bland flavored because they are so young. That’s why you can fry them or grill them without them becoming too tough. But your chickens are not going to pack on meat like those broiler hybrids. Some people butcher at 12 weeks old but there is just not much meat there. But at least they can fry or grill. I like to wait until 18 to 22 weeks. There is a lot more meat and somewhere in that age range they slow down on packing on the weight. You can eat any chicken, regardless of age or sex, but you have to alter your cooking methods to make that work. The older they are the slower they need to cook and the more moisture they need to keep them from drying out.
That is a lot of chickens. How much freezer space do you have and how much chicken can you eat? That might factor into it too.
One reason some people butcher cockerels pretty early is that they go through puberty and get pretty rowdy. Puberty might start pretty young, maybe even before 12 weeks, or some will wait until 20 weeks or more to start. And the older they get the more space they need. With that many cockerels and that few pullets, I’d suggest you separate the pullets by 12 weeks to make their life a lot easier. The sex links should be sexed pretty well but there is a fair chance you will have a pullet or two in with your Barred Rocks. You might look out for that and separate any of those pullets too. You can eat the pullets but with that many cockerels hitting puberty they are likely to be really harassed.
When to butcher dual purpose cockerels is not a clear cut question. It’s going to depend on your circumstances. I’ve waited until 7 months when my freezer was full and adjusted my cooking methods accordingly.