"Let us finish this rivalry."
Translation:Ayo selesaikan persaingan ini.
Good question, David31008. The reason why /-kan/ is added to the end of /selesai/ here is because /-kan/ is a Causative (or Benefactive, in other cases) suffix.
Essentially, the speaker is saying, "Let's cause (or make) this competition (or rivalry) to finish." In this case, the idea is that the competition/rivalry isn't going to end itself. Rather, it must be intentionally ended by the agents/participants involved in the competition (or perhaps even by people outside the competition who would step in for the purpose of intervening in it).
1) The "us" isn't explicitly stated. Rather, it is inferred from the use of "ayo" immediately preceding the verb. Using "ayo" here is closer to a command or imperative. You can say "Ayo pergi" to mean 'Let's go' (or also for 'go away' too, I suppose).
2) Yes, "mari" can be substituted with "ayo." In that case, it might be additionally helpful to include the subject pronoun between "mari" and the "verb," but maybe not necessary.