If it's a big clump, I wouldn't worry about compromising the increases. The remaining mature rhizomes should produce plenty of replacements. With the smaller clumps, you should probably wait until the increases have gotten close to a mature size. By then, root systems of the spent rhizomes are pretty much gone (once the fan of leaves on top of a rhizome dies, the root system starts to fade away as well) and they are relatively easy to pop out of the ground after they are severed from the others.
It might be heresy to say it, but other than planting newly acquired rhizomes, I really don't mess with my irises much during the summer. I like to do most of my thinning either late in the fall after a couple of killing frosts or early in the spring right as they start growing. That's when it's easiest to see what is going on, it's more pleasant weather-wise, and there isn't too much concern about rot. I'm sure it's different in other places, but rot is really only a potential issue here from late spring to mid-summer. So, I tend to leave them alone during that time.
As far how to nurture them, I follow the practice of benign neglect with the occasional kick in the rear (thinning or dividing). Keep the beds relatively weed free, replenish the soil now and then, and prevent them from getting too crowded. They really don't require much more care than that unless you live where borers occur or the conditions are really favorable for leaf spot. I suspect more irises are loved to death than die of neglect.
