This is what I do when I hike alone:
1) Leave a detailed itinerary with someone you trust including a time by which you will contact them, and the emergency number that that person should call if they don't hear from you at the appointed time. Stick to the itinerary so any rescuers can more easily find you, don't deviate.
2) You answered your second doubt with the paragraph following it. Be assured that the farther you get away from roads and civilized areas, the more likely it is that any folks you do run into will be like-minded individuals bent only on enjoying nature.
As for the other questions:
Has anyone ever been discovered and troubled by other people?
I've met some troubling people when out, but not in the criminal sense... I once had the pleasure of meeting a lovely elderly woman who had been hiking most of her 85 plus years, and had a penchant for nude swimming. I managed to be on the other side of the lake while she partook of the waters.
Anyone ever got ill while they were out alone?
No. When hiking with a group; I've been ill, and had others around me become ill or injured, though. Just be prepared, and have fun. It's not all as dark and foreboding as some would like us to think.
Enjoy!
Josh