You should call the PCB fabricator you intend to use, and ask them your questions. Different fabricators have different capabilities depending on their equipment and experience.
In general, an odd number of layers is not a good idea. Because of the unequal distribution of materials, the board will tend to warp and twist.
There are fabricators who can handle 10mil finished hole size on a 200mil board. Again, your best option is to call your fab and ask them.