I've had a variety of achilles/heel issues over the past 9 months, and stretching is definitely something that PTs and orthopedists will tell you to do. My guess is that by stretching it you take strain off of it and allow it to heal. The things that have been really stressed for me are stretching both the calf and the achilles, basically a bent leg calf stretch. This may do more to stretch the muscles around the achilles than the achilles itself, along the lines of the previous poster that is very important that the muscles around the achilles be flexible, so hamstrings as well since the back of the legs are all connected in some way. Just make sure none of the stretching hurts, it's all got to be very gentle with an injury
In terms of running, my injury issues started from something really minor that turned into something very major from not dealing with it initially and continuing to run. If it hurts to run with shoes on, something isn't right and you should probably rest it. It sucks when it happens, but it is way better to take a week off of running than 4 months.
Good luck with it.