I have some mixed breed chickens, mostly stemming from Rhode Island Reds and Barred Rocks. They average and egg a day during the warmer months, production slows in the winter but can be kept up if you provide light and some minimal warmth. They are very hardy and will breeze through winter. So if you get half dozen layers then you'll have a great supply. Extra eggs can be sold to neighbors to cover feed cost. Around 50 lb. of layer mash will last around 1.5 months for half dozen layers.
They begin laying in about 6 months on average and continue for several years, this is breed dependent. The "Black" or "Red Sex Link" chickens are reputably very good layers, but many breeds will not disappoint.
Care is minimal. You 'should' have around 3 sq ft of floor space per hen, but if you allow them some range area this can be drastically reduced. They don't like snow and most times won't go out in it. Consider in your area, I know it is small, a small hen house or even a Chicken tractor. If you made a nice one, it should compliment your yard. I can't recall off the top of my head how big your yard is (from the pics).
I will say, it is some of the most entertaining way to spend your time watching them. They are really funny. You can also feed them just about anything you'd normally compost...oh and spent grains they go nuts. If you keep your spent grains cool you can easily feed them for a good while on that.
Water is crucial, but you can buy double walled containers in sizes of 2.5/3/5 gallons. They are great, a 5 gallon will keep my chickens with water for over a week on a filling. As for food, you can hang a feeder as well. So you can easily stay away for several days without worry, the only problem is the egg collection which you want to keep up with. Perhaps a neighbor can be coaxed into doing this in exchange for eggs collected. The main reason being you don't want your hens to get broody, but also the eggs can spoil....a big however here...fresh eggs shouldn't be washed, they have a protective layer that will keep them for many days out at room temp (actually unless it is very hot, we keep a dozen on the counter all the time).