Why are RG flow fixed points associated with different phases? I thought the RG makes only statements about behavior near to critical points... a definite phase is far away from the critical point, right?
RG fixed points themselves are not associated with phases. Rather, RG fixed points (and their basins of attraction) describe systems at a phase transition. You are correct that taking the limit as a system approaches the critical surface only allows us to make statements about the system near the transition, but the RG is also used more broadly. The different relevant perturbations of a fixed point describe the possible phases (low-energy behaviors). By tuning the relevant perturbations to zero, we reach criticality, e.g. phase transitions.