AllRight:
1. Will admissions committees really trust my second recommendation? (I mean the conflict of interest due to opinion from other cofounder)
A: Yes, trust is not really the core issue here.
2. Is it enough to explain that the co-founder is the chief executive and I am his deputy and another co-founder, so that there is no doubt?
A: Yes, just be clear about the relationship. If the relationship is really 1a and 1b, that's fine. Even if it were 1a and 1a that would also be fine. Not everyone has an opportunity to get a second superior to write a letter of recommendation. Sometimes founders have to ask customers. Don't worry about the relationship as much as the clarity of the role.
3. We are a non-profit organization and the title co-founder does not play such a role as in the business environment. So, I can simply avoid mention the fact that that I am a co-founder at all.
A: Why? I think you're looking way into this matter. What matters here is the core content of who you are, how you operate, and how you are successful. All this title, hierarchy, etc. is just context so the AdCom understands who is writing the letter what is their relationship to you.