Adding classic tween breaks animation and moves assets

I would almost say make sure every part you are trying to tween is a Graphic symbol. better to tween that way than tweening objects. The reason I say this is I am looking at your library and you don't have any graphic symbols. Right-click each part of the bear and put it into a graphic symbol.  Just how I a perceiving what you are asking help for. 

there's a key concept you may be missing:

1. you probably want to animate the bear (with it's arms/legs etc) across the stage.

2. you probably want to animate the bear's arms and legs etc so it looks like it's walking.

to do both, create a movieclip or graphic symbol of the bear.

you can tween that symbol across the stage.

in the bear's timeline, animate the arms and legs etc moving in a walking motion.  to do that, each bear body part should be in its own layer.  animate each body part. 

it's at this point where you want to adjust the transform points so that, for example the arms move from the shoulder joints and the legs move from the hip joints. 

to get more realism the legs and arms should be movieclips or graphic symbols so they can animate as the entire appendage swings.  ie, the knee and elbow joints will swing too (and you will probably need to adjust your transform points of the forearm and shins to the elbow and knee.)

You might check your symbols to see if the transformation points got moved. Select each symbol with the Free Transform Tool to see where the transformation point is. By default, it would be in the center of the symbol. Here's more information on Transformation Points:

• Move, realign, change, and track the transformation point