If I miss my connection and have to wait until the next day for a new flight, and either there is no option to go to a hotel or I decide not to, will the airline provide me with a cot and blanket, food and water?

This depends on both the airport and the reason you are spending the night. Some, but not all, airports have a limited supply of cots, pillow, and blankets, as well as basic toiletries and food coupons that are most commonly brought out when passengers are stranded due to a major weather event. If you just miss your connection for some other reason and decide to remain at the airport overnight expect to sleep on the floor or across some chairs and provide for your own needs. There are certainly exceptions, but these services cannot be expected.