Unequal Comparison Relations

It's one and the same thing whether you keep or omit "does" and "do" in your chosen sentences. There is no deep reason behind why it was done in one case and not in the other. To not let others bat further eyelids, it is suggested that you keep them or leave them out in both the cases.

The answer is that if you elide (that is omit) the word "does" in the first sentence it becomes a little ambiguous. If you say "He visits his family more frequently than she" you could take it to mean that he visits his family more often than he visits her, in spite of the fact that 'she' is the nominative form of the pronoun and 'her' is the accusative. If you were to say "He speaks Spanish more than she" that would, grammatically, be quite correct.

Having said that the elided form generally is much less common now than it used to be and tends to sound pretentious when it is used. It would be unlikely that you would hear "He speaks Spanish more frequently than she" or "He visits his family more frequently than I" in normal conversation. You would be much more likely to hear "He speaks Spanish more frequently than she does" or "He visits his family more frequently than I do"

When you are speaking I'd forget about the elided forms altogether and say the verb aloud.

Both are correct; but "than me" would also be correct.

Quora

• Both ["than me" and "than I"] are correct. Using “than I do” is slightly more formal, but they mean pretty much exactly the same thing. “Nobody does it better than I” is also acceptable.
  Strictly speaking, “better than me” is ambiguous, and a good long time ago there was indeed a significant distinction between “than I” and “than me” that eliminated that ambiguity. But in the modern language, that is no longer the case, and nobody really cares about the ambiguity that results because the alternative reading is semantically ridiculous.