The issues you are mentioning are unrelated to the post’s title " How to get the position of the front side of a part". What I provided above is the answer exactly to your initial post’s title and description, but now you seem to be asking about way more things, and I doubt anyone can answer all of these with so little context.
It would be too much to write code to fix all of these, but here are some tips I would give you:
This is because you tween to the initial monster’s position. However your monster seems to be moving even after the Tween is created. If your monster moves a little higher or lower, then the initial point you started tweening the camera to, is not the correct one anymore. It seems like you want more of a Camera Focus/Following system, rather than a Tween.
How do you control your monster’s movement? If you are using body movers, make sure to read on AlignOrientation.
Here there are many moving parts that could go wrong. How often are you checking for the distance between the monster and player? How do you actually make this check? Are your scripts local side or server side?