How much damage they will do to your lawn depends on chicken density, your climate, and somewhat on your seasons. They do more damage when plants are not growing than when they are. I don't know how big your back yard is but it's unlikely four hens will destroy that much. They will dig a hole or two for to take dust baths. Often the area right at their coop will get cleared. It's unlikely the damage to the grass will be extensive if your yard is as big as I think with only four hens but expect a few bare areas, especially where they like to hang out.
Chickens can destroy areas in different ways. When chicken poop builds up and does not have time to break down the concentration can burn plants. It kills plants by having too much nitrogen in that area. Their walking can compact the ground and make it hard for stuff to grow. The big problem is that they eat about anything green and then scratch up the roots and probably eat them. Anything that sprouts gets immediately eaten.
There are three basic possibilities. If they are confined in a small space they will make that into a barren waste. In a larger area they will eat the plants that they most enjoy and leave the plants they don't, usually some ugly weeds. It may stay pretty green but that can become a jungle. You'll probably want to keep it mowed for appearances plus to give the good stuff a chance to grow instead of getting crowded out. The one I'd expect you to see if you have the area I envision would be a few isolated areas of damage but the main lawn not destroyed at all.
If the poop is spread out it is unlikely to stink at all. If it builds up and gets wet it can stink. Smell is highly unlikely to be an issue with you except for the coop or if you leave hen locked in a small run most of the time. You can manage that by keeping it dry.
There are some possible issues. As Micheal mentioned, they love to scratch in flower beds or anything with mulch. They can do damage to landscaping beds with their scratching. Don't even think of having a vegetable garden unless it is fenced against them. They will eat blueberries, blackberries, grapes, and things like that. It is hilarious to watch one stand under a grape vine and jump a couple of feet to snag a grape.
Wherever they go they poop. A lot of the time they like to perch on lawn furniture or on your back porch or patio, especially if it is covered so they are in shade. You are likely to find their little land mines everywhere. That may reduce your pleasure in using your back yard for entertaining or just for your family, especially if you have small kids.
Lots of people let chickens roam their back yard and really enjoy it. They either put up with it or they fence the areas they don't want the chickens to damage or pollute.
I think it is an excellent question, at least you are thinking. And welcome to the forum, glad you are here.