There are a number of factors to consider when you are picking out a running shoe. You are spot on to consider your foot and how it is shaped and how you carry your weight around, yet you should also think about your particular running style and your leg stride (i.e. how your feet and legs land on the ground and propel you forward during a run). All of these factors are important to take into consideration, because they play a role in determining how much cushioning your shoe needs, what kind of support you need, how high the ankle of your shoe rises up, what material they are made out of, and how wide or narrow they are. Flat feet is a common problem to have as a runner and, fortunately for you, the right shoe can fix most of your troubles.
Flat feet can cause a lot of pain. The foot's arch has a tendon in it shaped like a bow, that functions to absorb shock (especially the high impact shock that comes with repeatedly planting your feet and bounding off of them during a run). When you have fallen arches, though, that tendon does not absorb as much shock and can cause a great deal of pain that can travel through your foot, to the heel and up into your lower leg and calf. In the long run, it can also throw off your posture.
Luckily for you, our website actually has an entire shoe review of the "Best Running Shoes for Flat Feet," found here. This list of shoes all contain shoes with a great deal of cushioning in the sole and support. oftentimes, they will come equipped with nifty and comfy gel inserts that can serve as a kind of substitute for the tendon that traditionally absorbs shock. These shoes are not minimalist shoes, but rather, run wider and with more material, padding, and fabric surrounding the shoe. They also have a higher rise in the ankle and on the back of the shoe, to help prevent over and under pronation of the ankles (i.e. turning inwards or outwards, causing strain on the insides and outsides of the feet and ankles).