I recommend you start with a harmonica in the key of C for several reasons. Most importantly, a C harmonica will enable you to be in tune with the majority of beginning level harmonica lessons.

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).

Yes, but you can speed up the process a lot by learning from others. There are many great harmonica tutorials on youtube, for example. Different teachers work on different styles of playing and different genres of music, but the most common is blues harmonica.