There are several questions in the OP. The answers to all of them depend on the lease terms.
If Person A wants out of the lease, is the only option to just try to negotiate a way out of their portion of the lease with the landlord, presumably by payment or other means?
Yes, usually. Regardless of who is on the lease, Person A is on the lease. In order to terminate or break the lease, Person A will have to negotiate termination. Most residential leases provide for joint and several liability for lease obligations. That means the co-signed lease is like three non-exclusive leases, one each between landlord and Persons A, B, and C. So Person A has to find a way to break the Person A lease. Other than a breach by either party, that likely can only be done by negotiation with landlord.
What rights do Person B and C have if Person A does stop paying?
Absent some other relationship or understanding between them (that is, other than the lease,) likely none. Persons B and C are each fully responsible to pay all of the rent. In other words, as far as the landlord is concerned it does not matter who pays the rent as long as it gets paid. If it doesn't get paid, the landlord can evict and sue all 3 for non-payment of rent. But the lease likely does not discuss the relationships between A, B, and C - whether they pay pro rata by time in the unit, by space used, per capita, or whatever. Landlord doesn't care, and is not the counterparty to those decisions.
That said, if there is a relationship between A, B, & C (for instance, if A & B had a contract describing who would pay what, and B entered into another contract with C,) that will determine their relative obligations.
The landlord agrees to allow the lease to be amended for another person, person C to be on the lease. Person B negotiated this with the landlord without consent of person A.
This may create liability between B to A, C to A, or B & C to A. It is even conceivable that it creates a liability from landlord to A, if A had a reasonable expectation that the lease would not be amended absent A's consent. And the amendment may not be enforceable against A. So, for instance, A may be able to kick C out of the unit and bar C from reentry. It will not effect A's liability to pay rent.