If you're not sure about the harmonica type, start with a standard 10-hole diatonic. You can't go wrong with the Lee Oskar Major Diatonic or the Hohner Special 20 (preferably in the key of "C"), both are great for players just starting out.

With regular deliberate practice, you can expect to be playing simple pop tunes within around 3 months. Within 6 to 12 months, your technique will improve and you will probably be able to work on bending notes (a very important skill for getting the best out of a harmonica).

There is no difference. A harmonica is a mouth organ. There is another instrument called a glass harmonica that is not a mouth organ, but a musical instrument consisting of a series of rotating glass bowls of differing sizes played by touching the dampened edges with a finger.