I had a sim autonomously play with the training dummy before she was skilled, but she did have the disciplined trait. As for fixing things, I had a butler kill herself trying to fix a broken teleportation pad while singed. That might have been because she was a butler, though, and they're supposed to do repairs. Generally my sims just roll wishes to fix broken objects rather than actually doing it. (Procrastinators!)
I've never seen a sim read a skill book autonomously. Mine will automatically play chess either on the computer or chess table often. I'm not sure if this is before or after having a skill point in logic, though. I've seen artistic sims autonomously paint, and use the drawing or sculpting table, even if they have no points in the skill. The same thing seems to happen with virtuoso sims and musical instruments.
I've had some sims do laundry on their own, but usually only if they're neat and/or the clothes have overfilled all the hampers in the house. My sims will often do their own dishes after I've made it a habit for them, but only neat or perfectionist sims seem to do that on their own automatically with no forcing the habit on them.
I've never had any sims teach toddlers skills autonomously, even family-oriented ones. At most they just roll the wishes for it.