There is no one set distance, it depends on how big you want the puppet to be in the frame. Usually you use a combination of wide shots, medium shots, and close-ups to tell your story.
The wide shot gives a sense of the environment, where the puppet is. The closeup focusses on one particular character, and lets the viewer see fine detail. So naturally you would move your camera back a bit to get a wide shot. You might also use a wide angle lens.
For a closeup you usually move in closer. You don't want to be closer than your lens can focus, obviously. I use a 55mm macro, that can focus up close, so I might have the front of the lens only 200 - 300mm (8 to 12 inches) from the puppet for a close shot. For a medium shot with the same lens I might be 600mm (2 ft) away, or more. Or I might change to my 28mm lens and not move as far back, because the wider angle of view will show more of the sides of the set as if I was in the room, instead of seeing it from a distance through a door.
For some wide shots I could be up to 2 metres away, if I have the space to move back that far.
You need light that does what you want it to do - doesn't have to be professional lighting, it's where you put the light. A basic setup is called 3-point lighting. You have (1) a key light, that's the brightest, maybe up above and to one side. If you were outdoors, that would be the sun. You have (2) a fill light, softer and less bright - it might be a white board to reflect the key light back at the puppet, or a different light. It goes on the opposite side and from the front. That's to fill in the shadows so they aren't too black. And you might have (3) a backlight, or rim light, that is above and behind the puppet, that brightens up the edges and helps it to stand out from the background a bit. Here's a diagram: http://www.mediacollege.com/lighting/three-point/ There are a few tutorials on YouTube to show how to set up, and the effects of each light, just google 3-point lighting and you'll find them. I use lights bought at disco lighting stores mostly, but will also grab a desk lamp with a compact flouro coil in it if that puts the light in where I want it, and the right colour of light. What's right depends on the effect you want - sometimes I want a hard light with black shadows, like a Film Noir look, but at the moment I am shooting 2 bird puppets on a branch that need to match the background footage of a cloudy day with soft light that doesn't seem to come from any direction. For that I bounced my fluoro lights off the white ceiling, and left on some of the long fluoro tubes on the ceiling that I have for work lights, to get a soft ambient light with no real shadows, and only a hint of brighter light coming from above. So there is no one answer to what is "good light".
I think tie downs are essential, even if the puppet is not walking around. Well, something to hold the puppet steady so it doesn't jiggle around. I only use rigs if the puppet has to fly, jump, or run, any time both feet will be off the ground. A sitting puppet could be anchored by a screw in the bum instead of the feet, or held by a flying rig I guess, but since all my puppets have tiedowns in the feet, I use those.