Would like to know what bebegim translates to....and also if it can be used in different ways in a conversation depending on relationship

Bebeğim means "my baby" . To use of it depends on you. 

For context...i have a very good relationship with my gf, but she uses that word with her ex bf. Would that be inappropriate?

I wouldn´t say inappropriate. But she should be choosing another word, put it that way.

It translates to ´honey´, ´baby´, ´sweetheart´.

I´m sorry D, but if your wife/girlfriend would use that to her ex, would you appreciate that vey much? No.

My boyfriend would have serious trouble if he said that to his ex. No, no, no, non, non, non, hayır, hayır, hayır.

...Yes, you´re right.

same here! if your g/f is native turkish, so it means there is a BİG problem for me! she could not say anything like that except me!