The Internet consists of economically selfish players in terms of access/transit connection and content distribution. Such selfish behaviors often lead to techno-economic inefficiencies, such as unstable peering and revenue imbalance. Recent research results suggest that cooperation-based fair revenue sharing, i.e., multi-level Internet service provider (ISP) settlements, can be a candidate solution to avoid unfair revenue share. However, it has been under-explored whether selfish ISPs actually cooperate or not (often referred to as the stability of coalition), because they may partially cooperate or even do not cooperate, depending on how much revenue is distributed to each individual ISP. In this paper, we study this stability of coalition in the Internet, where our aim is to investigate the conditions under which ISPs cooperate under different regimes on the traffic demand and network bandwidth. We first consider the under-demanded regime, i.e., network bandwidth exceeds traffic demand, where revenue sharing based on Shapley value leads ISPs to entirely cooperate, i.e., stability of the grand coalition. Next, we consider the over-demanded regime, i.e., traffic demand exceeds network bandwidth, where there may exist some ISPs who deviate from the grand coalition. In particular, this deviation depends on how users' traffic is handled inside the network, for which we consider three traffic scheduling policies having various degrees of content-value preference. We analytically compare those three scheduling policies in terms of network neutrality, and stability of cooperation that provides useful implications on when and how multi-level ISP settlements help and how the Internet should be operated for stable peering and revenue balance among ISPs.