Yes it is wrong actually, lära sig just means learn and if you add that they teach themselves, you're actually adding something that wasn't there in the Swedish sentence. There's nothing in that sentence to make you infer that they teach themselves. The reflexive pronoun just expresses the fact that the language in itself is aware of the active nature of learning something.
Translation in practice is another story, there may be contexts where – in English – The children teach themselves to read means exactly the same as The children are learning to read and in those cases of course it would work.
Yes, that also works. The important thing is that lära almost always either have to have an object as in your example lär barnen or be reflexive jag lär mig. Then it can mean either to gain knowledge for yourself or to give knowledge to others.
So you can’t say e.g. jag lär tyska to mean ’I learn German’ or ’I teach German’. It has to be jag lär mig tyska and if you teach as a profession and don’t have an object you say jag undervisar i tyska, or jag lär klassen/barnen tyska but then it sounds like something temporary rather than a profession.