I agree that you might need to do some strength training, but would avoid deads and squats whilst your back is injured - I am speaking from personal experience. If your back is injured then your form will be off and you might make it even worse.
Time is the greatest healer when it comes to back injuries and it sounds like you have been given the right advice and are on the right course of treatment.
I missed three races due to a lower back injury, so have sympathy for your situation. It will get better though.
I didn't find out that I had asthma until I started running at the age of 26, one of the causes being exercise - it sounds like you have the same problem. I would recommend going to the doctors. You might need to go on daily preventative inhaler. Once you are on the right course of treatment you will be okay to continue with your running. You might have some form of viral infection if you're feeling drained of energy, definitely worth a trip to the docs.
I think that you're covering the right distance to do a 10K and that you just need to try and run for a longer period of time without stopping, You don't say how often you are going out for a run. My advice would be to go out at least three times a week and try and cut out some of the time spent walking. You could also mix it up a bit by replacing one of your sessions with two miles of hills to build up some more endurance. It might also be worth getting involved in one of the wekly parruns, which are a times 5k race.