@Oldřich Spáčil. Thank you for the comment. I know Lagrangian intersection is not a topological issue. What I meant by classes was modulo Hamiltonian isotopy, not modulo homologous manifolds, so that self-intersection of nondisplaceable Lagrangian is not zero.

Thank you. I should've thought about it carefully before I post the question. For the second question, my naive hope was to find suitable classes to do intersection theory. It seems to me that people think about Lagrangian intersections, but intersection of two Lagrangians is not Lagrangian. So I was curious if we can consider larger classes modulo Hamiltonian isotopy containing Lagrangians. I have no idea whether this makes sense at all due to my ignorance.

I hope so, but for example see Theorem 2.15 in page 24 in the link. Proper immersion is assumed and they didn't mention about branched minimal suraces. ugr.es/~jperez/papers/bamsJan11.pdf

Isolated singularities should not matter as you said, but I need some time to convince myself. Thank you.

So do you allow minimal surfaces have finitely many singular points? I want to be sure that the Monotonicity lemma still holds in that case.