With some justifiable distortion, one can claim that the Indian Adhyatmic traditions revolve partially around some ‘philosophical’ questions: what is real and what is existence? Is there a distinction between the two and, if yes, how to draw that line? This is a distortion because the debate never took this form; it is justifiable nonetheless to present it this way because it gives us a handle to tackle them today. However, we need to note that this is our way of posing the question, not theirs. Therefore, whether or not “believing” in Rama or Krishna entails the existence of such entities requires clarity about notions like ’existence’ and ‘belief’ in the Indian traditions. I do not have this clarity and, only recently, I was told that the Indian intellectual traditions (in Sanskrit) do not have a word for ‘belief’. (It is as yet unclear to me what this implies: does it have the same force as the statement that Indians did not have a word for ‘gravitation’ either or whether the notion of knowledge that we have is different from what we think knowledge is today.) Consequently, I cannot say much about this discussion today.
Only recently, I have begun to come to grips with what meditation is (as I can endorse it within the framework of my knowledge about the 21st century world). This understanding of what ‘adhyatma’ is has also helped me come to grips with the notions of ‘real’ and ’existence’, the way, I believe, Indian traditions have used them so far. One of these days (hopefully, soon) I intend to write a note on the issue of adhyatma. Until such stage, I am afraid you have to be content with this (rather cryptic) remark.
You ask: “What does it mean to be a bhakta of Rama or Krishna? Does it mean we should believe certain things about them? Or does it mean we display certain attitudes towards these characters and stories?” The answer is: choose whichever option helps you to become their bhakta, if that is what you want. To bhakti, both the questions and their answers are utterly immaterial. Ramayana is a way of achieving enlightenment and neither the success nor its efficacy depends on either the questions or their answers. Answering these questions will not help you become their bhakta; neither does ignorance about these prevent you from becoming one.
In the simplest possible terms, and to repeat once again: Ramayana is a way (to some, the only way) to reach enlightenment. When I write the note I intend, you will see that this is the basic truth about Ramayana. (When you understand this, as our traditions never tire of saying, you will have understood everything.)