Running to alleviate back pain?

I'd suggest biking instead -try a spin class or go outside for a good long or quick ride. Running to eliminate pain from fallen arches is counter-productive. Try another exercise until some of the weight comes off.

I have found alternating run with walk as my body can handle it best. It even strengthened my legs to support my bad knees.

Go to a running store and tell them your goals and pains. They can help you find the best shoes for you as they watch you walk. The shoes are worth the price.

I have also read that alternateing run with walk is more like a fitness circuit and better for you.

Before I'd consider running I'd suggest walking and some strength training (which could be body weight exercises) to strengthen your core, hips & glutes.

Once you can walk comfortably for half an hour or so you could start adding short, slow running intervals (eg 1 min run, 9 min walk - repeat 3 or more times) and gradually shift to 2 min run, 8 min walk etc until you can run 10 min continuously with a 1 min walk between intervals at which point you can decide whether you want to run longer continuously or not) Once you've built up your aerobic base (important for cardiovascular health) you may want to start doing some speed work but unless you planned on racing you could probably skip that.

On non-running days you may find some short HIIT workouts beneficial (tons of them on youtube) but don't be surprised if you have to keep them really short at first, if you're really dong HIIT a brief workout can be challenging and exhausting.