I would say that yes, they are acceptable. For example, a kid talking to his mom:
Parent: Did you do your homework?
Child: Yes I did!
Parent: No, I don't believe you.
Child: I did do it!
As Subjunctive mentions in the comments, another option is "I really did it!" Or you could combine the two - "I really did do it!" or "I really do care about you!"
It's used as a reply to disbelief. Another example for the third person.
Person A: I heard that Johnny does Taekwondo!
Person B: I don't think he does it...
Person A: He does do Taekwondo! I saw him at practice with my own eyes!
As for having something like "do do," read from Stangdon's comment below:
It's grammatically correct, but it sounds a little funny. "I don't do a lot for the company." "No, you do do a lot for the company!" We would probably try and find a better way to phrase it, like "You actually do a lot" or "You really do a lot".
I don't think that it is strange enough to say that it wouldn't be used at all, but "do do" sounds a bit funny to the ear. You can still use it, but know that there are other options.