Re: Difference between I can't speak German and I do not speak German?
For the first, it's possible that you know how to speak German and choose not to with "don't." With "can't" you lack the ability, even if if you wanted to. The "can't" version is more natural to my ears.
For the remember pair, there is no practical different in meaning. A person cannot choose to remember (let's leave the psychology of repressed memories out of this), so the possibility of being able to but choosing not to doesn't exist.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.