After a few 1km runs.... Well, we are all different but I NEVER run more than twice a week, so the body has plenty of time to recover and I NEVER run on consecutive days, because that is injury guaranteed for me. I might as an exception run 48 hours apart and then go back to the normal 3 or 4 days apart, but that is my best offer. That gives the body more time to recover. I have the same rule for swimming, and other activities like painting and decorating or difficult gardening. I learnt that even before I knew I had HEDS. Some people around here I read do go swimming more than twice a week, but whenever I have done that in the past, I have got a shoulder injury and have not been able to swim for six weeks. You need to work out for yourself how much your body can happily tolerate. I have even heard an interview by Paula Radcliffe saying much the same, except how far she can push her body before she gets injured is considerably more than me. We are all playing a balancing game, disabled and elite athletes alike.
For balancing on one leg, you could start with lightly holding on to something. I have to admit my balancing improved enormously but sort of cheating and having prolotherapy to tighten the ankle ligaments.
Lunging is a good exercise if done with good posture and alignment, probably with the help of mirrors at the gym to check.
Joint group leader for South Central (Bucks, Oxon, Herts)