Am I allowed to share a fee with an attorney that belongs to a different law firm?

Yes. The Code of Professional Responsibility allows fee sharing between lawyers not associated in a law firm under Disciplinary Rule section 2-107. However, the division of the legal fee must be (1) in proportion to the services performed by each lawyer or, by a writing given to the client, each lawyer assumes joint responsibility for the representation, and
(2) the client agrees to employment of the other lawyer after a full disclosure that a division of fees has been made.  Furthermore, the client must also be informed of the total share each lawyer will receive, the client’s agreement must be confirmed in writing; and the total fee must not be excessive.