I think that you're on the right track with doing yoga to stretch things out. You might want to try a foam roller, although it is quite painful starting out.
A lot of your issue could just be that you are new to running. It takes a while to build up the calf muscles necessary to run without experiencing that kind of muscle tightness.
Proper hydration and intake of protein is important too. Have a small snack and some water before you run, and then make sure to hydrate after you run.
You might want to try Gatorade, Powerade, or coconut water to replenish all the electrolytes that you use while running, and eat something high in protein to rebuild your muscles.
I'm with you on the forefoot strike, and that definitely is strenuous on the calves, but it eventually will give you great calf muscles.
I switched to wearing minimalist shoes a few years ago, and I'll never go back. Some people say that they cause injuries (and they do without proper preparation), but when I bought mine, I simply followed the instructions on the website where I purchased them (doing calf and foot strengthening exercise before beginning runs) and eased into running longer distances.
Aside from some initial calf soreness and blisters, they've been awesome. I love them!