Yes that is true.
I’m not too sure about that. I think it depends on which country you are applying from and which country you are a citizen of. In the Pakistani passport for example, the name of the father (for unmarried ladies) or the name of the husband (for married ones) is mentioned. Therefore you don’t need any additional proof, if your father/husband is the Mehram traveling with you. However if no Mehram is mentioned in your passport, then I think you need to submit a proof-document with the visa application. If you intend to travel with your son for example, that would be a birth certificate listing you as his mother.