Hi
if you're planning to move to Odessa I will be able to give you a piece of advice concerning your questions (though I'm not American:). If you want to purchase a life insurance, it's not a problem, and what is more, there are some respected international insurance companies already working in Ukraine (like AIG, etc.). If you plan to work in the sphere of health, I think it will be not very easy for you to find a job, as public hospitals don't need english speaking staff, and the salaries there are extremely low, and the conditions of work are unbearable. So you should look for a position in private hospitals. I think you should start searching right now, for instance publish you CV on some of the special rectuiting web-sites...
The cost of your living depends on your life style and the area where you're planning to stay. Of course, living in Nikolaev will be cheaper than in Odessa.
Should you have any further questions - don't hesitate to contact me
My primary business is apartment rental, but I'm always glad to give a piece of advice to someone moving or travelling to our city.