There is no magic ratio of hens to rooster that guarantees you no problems or that guarantee you problems. A single roo could be too much for one or two hens. Plenty of breeders keep one rooster penned with one or two hens and don't have problems. It purely depends on the personality of your individual hens and rooster. You are much more likely to have problems with adolescents, but if you can get through that age, they usually settle down. With that said, you can experience problems with any ratio, even 1 rooster and 20 hens. You are a little less likely to have problems with a few more hens than a few less hens, but no guarantees either way. It just depends on the individuals.
How much of a problem they are as far as chores is not related real close to numbers, at least up to a point. The daily chores should be feed, water, collect eggs, let them out of the coop in the mornng and lock them up at night. That is not really different for two or 15 chickens.
Where the difference comes in is more in poop management and possibly flock dynamics. If you squeeze them in a small space, you need to manage the poop a lot more regularly. If they have lots of space per chicken, you seldom have to manage the poop. You have less social in-flock issues with them if they have extra space too.
If you build a coop that is the minimum size and have a small minimum sized run where they spend their free time, you may wind up cleaning or raking almost daily. If you build something bigger, you may be able to never rake the run and clean the coop extremely rarely. Whether they free range or not on a regular basis also enters into the picture.
One potentially big area of poop management is under the roosts. They poop a lot at night. You may find it is less overall work to put a droppings board and collect the poop there on a regular basis, once a day, once every three days, once a week, whatever works out best for you. Or you can rake under there regularly, but you might have to clean the entire coop a little more often doing that. Instead of raking, maybe throw some scratch under there and let the chickens rake it for you.
You don't have to have a rooster for anything other than if you want to hatch eggs. Any other reason is personal choice.
There is a lot I don't know about your situation, such as how many eggs would be enough for you and your specific family. All four of you may have two eggs for breakfast each, deviled eggs for lunch, and you may bake a lot of cakes and custard pies. I just don't know. But I'd think a flock of maybe one rooster and 5 or 6 hens would probably be OK for most people in your situation. A couple of extra hens would not really make the chores that much more difficult as long as you have the room.
As far as coop and run size, I suggest you decide on the number of chickens you want and build something at least twice as big as the normal recommendations on here, more would be better, even if you plan to free range them most of the time. Usually the minimum recommended is 4 square feet per chickne in the run with 10 square feet per chicken in the run. If one of your main criteria is daily chores, the more room you can give them, the less you normally have to do.
