A key motivation for protocols that employ explicit congestion control, like the Rate Control Protocol (RCP), is that they may allow the design of a fair, stable, low loss, low delay, and high utilization network. RCP belongs to a family of congestion control mechanisms called dual algorithms. Some of the important notions of fairness, in the class of dual algorithms are: i) Proportional fairness, ii) TCP fairness, and iii) Delay-based fairness. We consider dual congestion control models coupled with these three different notions of fairness, and analyze the stability and the consequences associated with the loss of stability. We show that the TCP fair and the delay dual algorithms can undergo a sub-critical Hopf bifurcation, which is undesirable for engineering applications. In contrast, in the case of the Proportionally fair algorithm, we provide strong evidence to suggest that the occurrence of a super-critical Hopf bifurcation is what we should expect as the stability condition gets violated. Thus, from a design perspective, our analysis favors the use of Proportional fairness in the class of dual congestion control algorithms. RCP estimates its fair rate from two forms of feedback: rate mismatch and queue size. An important design question that remains open in RCP is whether the presence of queue size feedback is helpful, given the presence of rate mismatch feedback. We address this design question using tools from control and bifurcation theory. We study the impact of queue feedback by investigating the performance of RCP, both in the presence and absence of queue feedback. The key performance indicators that we consider are local stability, global stability, rate of convergence and the asymptotic orbital stability of the bifurcating limit cycles. We analyze these performance metrics using two different models that have been proposed in the literature. In essence, all our analytical insights reveal that the performance of the RCP can be improved by removing the queue size term from the protocol definition. RCP achieves short flow completion times at the cost of over-utilization and queuing when there are sudden changes in network traffic. In the literature, an admission management process was proposed for RCP to address this issue. In our work, we analyze the robustness of the proposed algorithm against various flow arrival and departure patterns. Using packet-level simulations, we show that the proposed process appears to be attractive for dealing with the admission of new flows while maintaining small queues. A consequence of being able to maintain stable and small queues is that buffers, in routers, may be dimensioned to be much smaller than the current, bandwidth-delay product, buffer sizing rule.