No. They have gonads.

Three examples of a sex-linked disease are; Hemophilia, Wiskott-Aldrich syndrome, and androgen insensitivity syndrome.

HIV, gonorrhea, and syphilis are examples of sexually transmitted diseases, which is something totally different.

Yes, it has been observed in many species like horses, dogs and cats. Those animals are born with apparent normal female genitalia, but fail to come into season as they dont have ovaries but testicles inside. It is very rare, and like in humans its an x linked disorder, so it is passed through the maternal line. However it is not called androgen insensitivity syndrom but testicular feminization or male pseudo-hermaphroditism.

That is just a good starting point. If you don't know what sex someone was supposed to be, then karyotyping can be a good place to start. It can also give a clue as to what underlying conditions there might be. For instance XY chromosomes and natural feminine features could indicate androgen insensitivity syndrome. On the other hand, androgynous features and an extra sex chromosome would indicate Kleinfelter's syndrome.

Excess androgen secretion may be related to polycystic ovarian syndrome, which affects one in twenty women. Another cause may be congenital adrenal hyperplasia.