Muay Thai is useful for self-defence at any stage of training. It's not like some other more arcane martial arts that only 'kick in' once you've done them for a decade or so. The principles of the guard, footwork, balance and covering up are useful right from the first lesson. That said, it takes quite a bit of sparring, conditioning and drilling for you to become comfortable (more comfortable, you're never completely at ease) in a situation where you're getting punched. There's no real way of saying in terms of months or years of training, since everyone trains at a different pace.
My advice is, train in Muay Thai, and you know you're on the right track. It's one of the most useful self-defence martial arts. Don't worry about when you'll be able to trash people on the street, since that's never a helpful mentality to be in. ALWAYS attempt to defuse the situation. NEVER go out on the street thinking 'Any guy tries to get my wallet, I can take him'. 20 years of Muay Thai won't do that for you.
In terms of being a pro, barring all the bureaucracy involved, you can start fighting after as little as a year, if you're fit, have previous experience, and train HARD. I don't know how many years most people train, but at my gym it's generally between 2 and 5 years of training to work up to your first pro fight. Amateur fights are much easier to come by, just consult your local gym. If you plan to train in Thailand, the managers of gyms love Farangs coming in and fighting, so it'll be very easy for you to get fights there, but only fight if you're ready.
Hope that helps
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