My daughters mother claimed our 5 month old on her taxes but we both financially take care of her can I claim her as well? without getting my daughters mother introuble?

No.  If all of you live together then only one of you can claim the child.  She cannot be a dependent on both of your returns.

And if the child only lives with the mother, the custodial parent is the one who can claim the child.

No, you cannot as only one person can claim the child.