Hi,
Yes, you can certainly go to Iwakuni or anywhere else in Japan by yourselves. It is not withing walking distance of Miyajima however. It is about 23 minutes by train from Miyajimaguchi Station.
Getting around is not difficult at all. Train stations have signs, machines and messages in English, and quite frequently in Korean and Chinese now as well.
It is true in general though that few Japanese can speak English with ease. In school the goal is rote memorization and passing tests, not learning to communicate. But that doesn't mean you are completely up the creek.
If you get hopelessly lost, it is better to find some older school or college age students and write out your questions in simple words. Japanese are still very poor at speaking English but are often glad to help you out if you lose your way.
Carrying a business card or matchbook with your hotel's address on it can be a big help in getting back. Taking a taxi is the ultimate quick solution, but they can be expensive for longer distances, especially if traveling alone.
Looking at your original itinerary, it's unclear what your interest in Nagoya is - among the big cities it is the least interesting. Unless you have something there of particular interest, you might consider skipping it. Visiting Nara near Kyoto for at least half a day would be well worth it though. Also Himeji Castle is close by and the finest in Japan, if you have half a day free for that, it is also very memorable. One other place you probably won't have enough time for is the Akiyoshido Limestone Cave which like Iwakuni is also in Yamaguchi Prefecture. It's a 300 million year old wonder with giant underground caverns and and river flowing past you. Something to try on a future trip.
Best of luck!