Shalom teabird. It is safe, and there are many ways to go to Bethlehem. If you want to go totally on your own, that's also possible by taking one of the blue & white shuttle buses from near Damascus Gate (be sure to ask which one goes to Bethlehem...) for less than 4 shekels each. Then, you will go through the checkpoint and have to make your own way to the center of town, though it is possible to get a taxi from the Bethlehem side, no problem.
Another way is to book a tour, including transportation, ahead of time. That can be done by hiring a local Jerusalem Arab taxi driver who has permission to enter Bethlehem and by contacting a wonderful Christian Arab guide who lives in Bethlehem. I'm not sure the forum will allow me to include those names here, so if you want me to help you with that, just write to me at suztours@gmail.com and I'll be delighted to set it all up for you, no charge.
About walking from Mt. of Olives towards Bethany - I'm afraid that the security fence makes that difficult to impossible at this time. Instead, you can enter Bethany/Al-Azzaria (to visit the Tomb of Lazarus) from the other side, but only with a taxi driver or other local person who has permission to do so.
It is safe for you to wander about the Mt. of Olives as long as you are careful. The thing is - there is a lot to see up there, so you will surely learn & see more and have a fuller experience by hiring a tour guide to take you. I do a 3-hour walking tour there, and again, would be delighted to help you. Let me know.
suzanne
suzanne pomeranz
SuzTours: Every Day an Adventure!
jerusalem, israel
suztours@gmail.comhttp://suztours.blogspot.com