Does using wrist weights count as strength training?

You are providing a little more resistance, so you would burn more calories because more energy would have to be expended for that movement... but that wouldn't count as strength training, in the class sense of it. It may help build up a little more endurance, but it's not really going to hit the muscles the way that actual strength training does. If you don't strength train, I TOTALLY recommended adding it in, because building some lean muscle mass does a body good

It depends on how you use them. If you simply wear them for running, or doing cardio, then it's likely going to be harder on your joints all the way up. But just your back? If it affects your back that comes from using bad form and just having generally weak core muscles (which makes you, overall, more prone to back injury).
Ankle weights are best used to wear during strength/endurance training and not cardio sessions. They provide excellent resistance if doing strengthening exercises such as donkey kicks, fire hydrants, scissors, lunge w/lateral raise, etc etc etc.
The general rule when using ANY weight is that you don't want to go to fast or be too bouncy... because that's going to put your joints in jeopardy AND that same movement can also strain the muscle.
When using weights, you want to control the movement on the way up and the way down... flexing through the motion the whole time (trying to avoid using momentum to "throw" them up, and so forth).