1. Set the camera to manual mode via the mode dial on top. This will give you full control of the exposure settings, etc.
2. Invest in a decent tripod. Besides helping with camera movements, it helps with long exposures and still shots.
3. Practice makes perfect. I'm not totally sure what you mean by "advanced," but rotations can be achieved by placing your set on a turntable. Other movements may require combinations of simpler movements (ie. sideways dolly + rotation + focus pull). Yet others, like shaky-cam require custom movements for each frame.

2nd place Darkness and Light entry: The Tree
Behind the Scenes

Check to make sure that the ISO is set to manual (probably between 100 and 800) as well as the white balance, and that any dynamic range optimization settings (like HDR, DRO, etc... ) are off.

“But the enterprise in which I’m about to embark on is fraught with imminent peril.”

HDR and DRO should be off. They normally are. However, you'd have to check your camera's manual or the web to figure out if your camera has that setting or if it's on.

As for your questions...

1. First off, Nikon zoom lenses, especially the electronic kind, are horrible for animation. Though, this isn't a light flicker thread, so I won't get into that. What seems to be the issue is the live view not displaying the correct brightness. This happens for me a lot. You just have to expirament with the shutter speed, aperture, and ISO to get the right brightness. Simply pressing the exposure button doesn't change the exposure. The little click-knob on the camera does that while you press the button. Of course, it could be a completely different issue than the solution I'm giving, so maybe a picture would help.

2. Like legoguy501 said, get a tripod. Make sure it's strong and heavy. Setting the camera on the table works, but I don't recommend it because you're always at the same angle. Build your sets modularly so that you can take them apart and re-arrange them to get camera angles. This way, your camera can remain off the table and you just move the set and lamps instead.

3. A lot of people simply slide their tripod on the floor frame-by-frame. That's how I do a lot of my stuff, but it's really hard without an animation program that has playback and onion skin. Their are some nifty camera sliders on Amazon that you can attach to your tripod. I've not used one, but I hear good things about the one I linked from other people.

Above all, practice makes perfect. It's hard to figure out the easiest way to animate things, but you just gotta experiment until you figure out what's right for you.

YouTube • Website

"Whatever you do, do all to the glory of God." - 1 Corinthians 10:31b

You are using a really weird choice of lens for that camera, I haven't been able to find much about it online but it does look like you can manually set the focus and aperture which is what is important (I have a thread about the best Nikon lens choices for stop motion although you might be able to get most of what you want out of this lens you might want to invest in a proper Macro lens (like the Nikkor 55mm f/3.5 micro) and maybe a wider angle lens, and all of the lenses I sugest will let you generally get a lot closer to your subject.

For fancy camera moves like controlled panning (vs dollying which the sliders Rio recomended) I love my Manfrotto 410 tripod head. It's pretty pricey for just a camera head but it is going to give you a lot of flexibility, plus every single person that has used mine went out and bought one afterwards. Shooting with DSLRs is pretty expensive as a lot of the equipment I couldn't live without is just pricey, but it can give amazing results.