First welcome to PPRUNE. It might be worth using the search function, as you will find some useful information on the questions you are asking.
In brief, I can give you some basic information on what you want to know.
1. You will find quite a few pilots that have qualifications in fields that are non aviation related, and you having a degree in medicine will not put you at any disadvantage. As long as you hold a professional pilot licence when you finish your training, that is all that is required.
2. Getting your first professional pilot job is difficult and very competitive. A small percentage of people get jobs straight away, but for many it can take years. There are numerous variables. At the moment there are more qualified pilots than jobs. There are a fair few experienced guys out of work.
3. If you obtain a JAR licence in the US, there is no conversion. However, if you were to get a FAA licence you would have to convert it.
4. There are numerous ways of gaining experience. You could instruct, parachute drop, aerial photography, air taxi, etc, etc. Sadly, you will find all this type of work is not very well paid, so depending on your life style will depend on whether this will be for you.
When you take on this task of wanting to be a professional pilot, you must think it through thoroughly, plan it thoroughly, be shrewd with your money and have a back up plan. So many people get into this without having a clear picture of what is involved, and end up disappointed and disheartened. On a positive note, if you have the passion for aviation and drive to succeed it is a great way to earn a living and very enjoyable.
I wish you luck, all the best.