Hi Labseb! We lost our power once today already for about 75 minutes and I'm hoping we keep it for the rest of the day. It got cold fast when we didn't have heat!
I'm no doctor, but it sounds like you might have a pinched nerve in your back. I have two of them myself due to areas of stenosis in my spine. The pain radiated down my hip and leg and all the way into my foot. The only thing that gave me relief was a series of three epidural steriod injections into the affected area. I urge you to check with your doctor to figure out what's going on and how to fix it.
As for exercise, almost every exercise engages your core (back and abs) so almost anything you do is going to affect your back and its nerves. I agree that you should avoid ab machines, especially that horrible twisting one that's considered to be one of the most dangerous machines in a gym. I put myself on a core strengthening program of low back and ab exercises and I think that's helped. Ab exercises done on the floor or on an exercise ball are better than the weighted machines. But my doctor told me to avoid leg lifts because they really strain the lower back.
You might benefit tremendously from physical therapy, both for your foot and your back. A PT can teach you how to exercise to strengthen your core and possibly help the PF in your foot. Is that something you've considered?
As for cardio without weight bearing, besides an arm bike (which you said you use), the only thing I can think of is a recumbent bike. Even though you push with your feet, it's not weight-bearing, so maybe it would work for you. Have you tried one?
Best of luck to you! I honestly think your next move should be to get your back checked out by a doctor and to consider physical therapy. Let us know how you're doing!