@Travis M. good for you for saving up money! I think it's the best way way to invest:
1) There is no right or wrong answer to this question, it depends what your appetite is? Do you plan to invest locally, or far away? Do you want to self manage or have a third party PM company manage it for you? Will you be partnering with anyone else on your deal(s), or will you go in solo (this will help determine how much you can leverage and what your buying power is)? After answering these questions you should get a better idea of what would work best for you.
2) You should definitely begin your search now and start analyzing deals asap. However , if you don't buy your first deal until a year from now that may not be the worst thing - because a lot of markets seem rather frothy now and waiting may not be terrible. But don't wait to start speaking to brokers and analyzing deal because the worst that will happen is that you will learn the markets, understand what makes a good deal, and learn more about what you want to invest in. The best case is that you may just find the perfect deal today!
3) FHA is not my specialty, so I will defer this to someone else.