Well, for set security, use clay or blue tack. That's what we use (clay). For cinematic shots, try using corner angles for more diversity, yes you will have to move your set. Also, close ups are good for cinematic shots. You should also try to work on set building and lighting to pull off a truly beautiful shot.

Well like we said, you don't need to be constantly moving your set, but when you want to change the angle, you're going to have to move your set.

What, pivot on the tripod axis? I can do that, yeah.
I just can't do more dynamic shots because I don't know how to get on top of the set. Like a little bridge for the camera.

build a custom stand out of legos perhaps?

YouTube uncompleted project: filming not in progress

Perhaps you detached your camera from the tripod?  OMG, has anyone ever done that before?!

With any project, planning needs to be done ahead of time.  You do blocking with your actors, human, animal or figurines.  You have your shots all planned out ahead of time so you know where you camera is going to be in the set and have your set built to accommodate that.

They didn't dig up a trench for Frank Oz to move Yoda around, they raised the set.

I'm pretty sure some Aardman sets are built in sections so you can take out areas for the camera to occupy.

And we've solved your movable set problem so you really only have one element to worry about and it's been answered.  Now go animate.

I find that plasticine does not work when it comes to securing sets. It melts and is too soft to start with. Blu-tack is much better.

If your camera doesn't support PTP(picture transfer protocol)/remote picture taking, use delayed picture taking to reduce bumping/shaking.

Rest it against the desk.