I have fallen arches and wear orthodics. I haven't regularly worn sandals or even low-top shoes in about eight years. A podiatrist gave me a series of exercises involving a tennis ball to strengthen my arches and those helped some. I am wondering if there are any other exercises I can do to help strengthen my arches. Conversely I am wondering if there are any exercises, e.g., squats, I should avoid.
I used to have the same problem, and everywhere I looked people and science suggested one crucial exercise; short foot.
Here is an article about it: http://blog.evidencebasedfitnessacademy.com/2012/10/03/evidence-based-exercise--short-foot.aspx
Another exercise usually prescribed is the toe curl, however, research has shown that the short foot exercise is more efficient at increasing the strength of the muscles supporting the arch. Here is a video showing both exercises: http://www.youtube.com/watch?v=8x1su-x0E6U
The main things you need to do are to increase intrinsic foot strength (by performing the short foot exercise) and to relax the muscles that are chronically strained as a compensatory mechanism for the dropped arch (foam rolling, or rolling a tennis ball under the foot).
The main problem with fallen arches is that the feet start pronating, and this causes the knees to go inwards when, for example, squatting. The risks here are a nonfavorable position of the joints in knees and hips, with an increased risk of injury. Unless you can maintain posture, so that your knees go parallel to your feet when squatting, I recommend you either lower the weight or skip squats until you starts seeing progress in foot strength. The same goes for deadlifts etc. and especially plyometrics such as drop jumps.
I really recommend watching the videos of "the gait guys", starting with this one: https://www.youtube.com/watch?v=TyRE9dReVTE
The above video includes an explanation of the anatomy of the foot, which muscles are involved in "holding the arch", and includes an exercise for strengthening these muscles, and learning to isolate them.
This is a bit anecdotal, but I have always had wide & nearly flat feet. I used to get pain in my arches, and for a while I required insoles and extra supports in my shoes. I started weightlifting and squatting, particularly I was taught to curl my toes as I squatted to keep weight on the heels. Over the course of a year or two, my foot pain went away completely, and I actually went down slightly in shoe sizes. At best its a correlation, but this correlated well with the time I spent squatting. Now I am able to wear Chuck Taylor's all the time, something that I never would have dreamed of years ago (narrow shoes, almost no padding).
In general, think about all the articulated movements you can do with your ankle / toes, and practice those with progressively increasing resistance.
In specific, one good arch exercise is to take something like pieces of newspaper or a towel, lay them under your feet, and scrunch them up into a ball using only your feet & toes.