Is there a chance of bone cyst spreading in other parts of the body, if it is present in some bone already?

No. This, however is true only if it is indeed a simple bone cyst. Other types of bone lesions that may be similar in appearance but are not true bone cysts might in some instances spread.
No. Bone cyst are a non harmful finding in the bone - that is very specific to that area. There are 3 types of bone cyst, these include simple bone cyst, aneurysmal bone cyst and traumatic bone cyst. A person can have multiple cyst in different bones- but it is not because they are "spreading".