How to separate body parts and make them move.?

There are some good intro videos around that are probably worth going through. One I did is here: https://extra-ordinary.tv/2018/10/22/project-wookie-a-beginner-youtube-playlist/ which shows every step I did. Its a few years old now so does not use the latest features, it shows EVERYTHING, so fast forwarding at times is recommended, but it has a series of episodes showing different steps I took.

Some of your questions (it can be helpful to list them here as I might forget some from the video sorry):

• To get an elbow, put sticks (bones) in the arms for the upper and lower arm segments. Leave a gap at the elbow, which causes the elbow to bend. I suspect you have sticks on one arm and not the other.
• To make the waist move, put a dragger on the waste. Click the body layer in the rigging panel, then there is an icon at the bottom whic is a dot with 4 arrows (left/right/up/down). Click that on the puppet near the waist and you will have a new dragger.
• The thing in the hand is caused by "triggers" and "swapsets". A trigger by itself by default hides a layer. Pressing the trigger key makes that layer visible. A swapset builds upon this. All the triggers in the swapset are linked. Exactly one trigger in the set will fire at a time, so triggering a layer will hide another layer. Those hand buttons at the bottom at trigger controls. Clicking one will hide the other one in the swap set. In the rigging panel, you can hide / reveal layers with the "eye" icon. Triggers and swapsets override what you set for visibility which is why you see it while rigging (you have the eye icon on) but don't see it by default when using the puppet (the swapset has a default trigger which is the empty hand).
• Arm joints are tricky. Independence for a layer is used when you don't want the movement of the child to affect the parent, and when you want to do hinge joints (rather than weld). Don't over do it!
• If the arm is not independent, positioning sticks is very important to get the bend right. I often put a stick from the top of the shoulder straight down the side of the puppet, so the arm does not warp the torso. I think put the stick in the arm positioned carefully down the center of the arm until movement looks natural. You just need to experiment.
• If the arm is independent, you need to have arm artwork and body artwork overlap. I normally go for the body going straight up from the waist, torso, etc up to the top of the shoulder. I then put a semicircle on the end of the arm so it overlaps the body and when it swivels you still get good overlap. Also, if the arm is a solid color, then you can put the arm in front of the torso or behind and it does not matter. If the arm material is textured, I find it works better if the arm is behind the body (lower in the rigging panel) so you don't see the semicircle at the end of the arm.

Note: I avoid independence unless required. I think people over use it. It is important and necessary sometimes, but for this puppet I think will could look better with the arm being dependent (no crown) and using sticks to control where the flexing and bending occurs.

I hope that gets you some progress. But I would read through more of the Adobe tutorials etc too - lots of good advice there that I cannot fit into a forum post response.