There are two people Jo and Bob. Jo is standing still and Bob is moving at half the speed of light. I believe time for Jo is 1.5 times faster than time for Bob, so Jo must see Bob moving at half the speed of light divided by 1.5. Is this correct?
You are making the mistake of assuming a preferred universal reference frame as soon as you say something like "Jo is standing still and Bob is moving at half the speed of light." Standing still relative to what? Moving at half the speed of light relative to what? Jo and Bob are in inertial reference frames and neither frame is the preferred frame. As such, both will report the same thing as far as the speed of the other one relative to their own frame.
Sure. Both parties will view the universe from the standpoint that they are stationary. I'm sure you've seen movies in which two people are skydiving and you are seeing things from the viewpoint of one of the skydivers. The other person deploys their chute and, from your viewpoint, you see them go sailing upward. Now think about how most kinds represent a chute opening with their hands when they are playing (and many adults, too). They will usually have their hand with a closed fist moving downward quickly. Then they will open there hand in the shape of a chute and, at the same time, move their hand sharply upward followed by moving it downward again slowly. But, of course, that's not what happens, is it? When someone deploys their chute they continue falling they just slow down. But from the viewpoint of someone else that is falling with them and who does not slow down, they look like the went upward because the person interprets what they are seeing in the context of the notion that they are stationary.
The reference is the surface of the earth. Jo is standing on the earth and Bob is moving across the earth at half the speed of light.
Relativity does not work that way. When describing a frame of reference it must be from the observers point of view, you are describing a 3 body problem. When talking about what Jo and Bob are seeing it must be from Jo or Bob's point of view, not the Earths. Just to make things interesting you are describing a circle, not linear speed, if they stay on the surface of the earth. It is important in these problems to set it up with as few variables as you can, if you are trying to understand basic concepts. Bob is the one undergoing acceleration, right?
At some point both started at the same reference point, otherwise there is no starting point to compare time passed. There is no universal now, only when you pick a reference is there something to compare to. Stephen Hawking's book "A Brief History of Time" is a good read on the subject. I highly recommend it.
I'm going to go out on a limb here. Why couldn't things be from the viewpoint of the Earth? For me, the Earth is a living sentient being, and therefore, could have a viewpoint in the matter.
Then it would be 3 body problem, and the math gets trickier. Every point of view is different, the reason it is called relativity is it is always relative to the observer. Change the observer, and the readings change. The other thing is when most people learn the math and how to think of classical relativity, we are talking about moving in a straight line. If you are orbiting the earth then again, the math changes a bit. Like most science you have to learn the simple math before you do the complex. I'm no expert, I was somewhat interested in relativity in my youth and learned a bit about it, a lot of which has faded over time.
It doesn't matter whether the Earth is sentient or not. The question asked about what Jo sees which means that we need to use Jo's reference frame. The question is poorly phrased because it says that Bob is moving at half the speed of light but it doesn't say in which reference frame. It also didn't give the direction vector of Bob relative to Jo, and that matters. Finally, I have no idea where the factor of 1.5 is coming from. I'm seeing an effect of more like 15%, not 50%.
I think you made a mistake with the 1.5 times faster; it's not 0.5 faster for 0.5 the speed of light. Jo's time should be only 1.15 faster than Bob's if his speed is 0.5 c faster.
Again, if you have 3 bodies from any one observer neither can exceed the speed of light. If you have two objects approaching earth at 0.5 C from opposite directions, then switch to one of the other objects point of view, the other object is approaching 7/8C, and you are closing on earth at 0.5. Any misunderstanding is a result of linear thinking, which doesn't work for relativity.